From MAILER-DAEMON Mon Apr 21 15:03:46 2008 Date: 21 Apr 2008 15:03:46 -0300 From: Mail System Internal Data Subject: DON'T DELETE THIS MESSAGE -- FOLDER INTERNAL DATA Message-ID: <1208801026@mta.ca> X-IMAP: 1204501626 0000000114 Status: RO This text is part of the internal format of your mail folder, and is not a real message. It is created automatically by the mail system software. If deleted, important folder data will be lost, and it will be re-created with the data reset to initial values. From rrosebru@mta.ca Sun Mar 2 19:42:10 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 02 Mar 2008 19:42:10 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JVxj6-0002is-Pz for categories-list@mta.ca; Sun, 02 Mar 2008 19:36:16 -0400 Date: Sun, 2 Mar 2008 18:36:33 -0000 (GMT) Subject: categories: Minimal abelian subcategory From: "Tom Leinster" To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 1 My colleague Walter Mazorchuk has the following question. Being abelian is a *property* of a category, not extra structure. Given an abelian category A, it therefore makes sense to define a subcategory o= f A to be an ABELIAN SUBCATEGORY if, considered as a category in its own right, it is abelian. Note that a priori, the inclusion need not preserv= e sums, kernels etc. Now let R be a ring and M an R-module. Is there a minimal abelian subcategory of Mod-R containing M? If so, is there a canonical way to describe it? Any thoughts or pointers to the literature would be welcome. Feel free t= o assume hypotheses on R (it might be a finite-dimensional algebra etc), or to answer the question for full subcategories only. Thanks, Tom From rrosebru@mta.ca Sun Mar 2 21:02:10 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 02 Mar 2008 21:02:10 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JVz0I-0001f3-6x for categories-list@mta.ca; Sun, 02 Mar 2008 20:58:06 -0400 MIME-version: 1.0 Content-transfer-encoding: 7BIT Content-type: TEXT/PLAIN; charset=US-ASCII; format=flowed Date: Sun, 02 Mar 2008 19:21:39 -0500 (EST) From: Joshua P Nichols-Barrer To: categories@mta.ca Subject: categories: Re: Minimal abelian subcategory Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 2 Hi Tom, Silly observation, but wouldn't the contractible category consisting only of M and its identity morphism constitute an abelian subcategory by this definition, albeit one that is trivial? It would seem that the question for full subcategories is more interesting (and harder). Best, Josh On Sun, 2 Mar 2008, Tom Leinster wrote: > My colleague Walter Mazorchuk has the following question. > > Being abelian is a *property* of a category, not extra structure. Given > an abelian category A, it therefore makes sense to define a subcategory of > A to be an ABELIAN SUBCATEGORY if, considered as a category in its own > right, it is abelian. Note that a priori, the inclusion need not preserve > sums, kernels etc. > > Now let R be a ring and M an R-module. Is there a minimal abelian > subcategory of Mod-R containing M? If so, is there a canonical way to > describe it? > > Any thoughts or pointers to the literature would be welcome. Feel free to > assume hypotheses on R (it might be a finite-dimensional algebra etc), or > to answer the question for full subcategories only. > > Thanks, > Tom > > > > > > From rrosebru@mta.ca Sun Mar 2 21:02:11 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 02 Mar 2008 21:02:11 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JVz1r-0001mG-Sg for categories-list@mta.ca; Sun, 02 Mar 2008 20:59:43 -0400 Date: Mon, 3 Mar 2008 00:27:05 -0000 (GMT) Subject: categories: Re: Minimal abelian subcategory From: "Tom Leinster" To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 3 A couple of people have pointed out to me - in private, I think - that th= e question has a trivial answer (namely, the subcategory consisting of just M and its identity map). Sorry. I probably misinterpreted what Walter said to me. Tom >> -----Original Message----- >> From: cat-dist@mta.ca [mailto:cat-dist@mta.ca] On Behalf Of >> Tom Leinster >> Sent: Monday, March 03, 2008 5:37 AM >> To: categories@mta.ca >> Subject: categories: Minimal abelian subcategory >> >> My colleague Walter Mazorchuk has the following question. >> >> Being abelian is a *property* of a category, not extra >> structure. Given an abelian category A, it therefore makes >> sense to define a subcategory of A to be an ABELIAN >> SUBCATEGORY if, considered as a category in its own right, it >> is abelian. Note that a priori, the inclusion need not >> preserve sums, kernels etc. >> >> Now let R be a ring and M an R-module. Is there a minimal >> abelian subcategory of Mod-R containing M? If so, is there a >> canonical way to describe it? >> >> Any thoughts or pointers to the literature would be welcome. >> Feel free to assume hypotheses on R (it might be a >> finite-dimensional algebra etc), or to answer the question >> for full subcategories only. >> >> Thanks, >> Tom >> >> >> >> >> >> > From rrosebru@mta.ca Sun Mar 2 21:02:11 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 02 Mar 2008 21:02:11 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JVz1E-0001jX-Kb for categories-list@mta.ca; Sun, 02 Mar 2008 20:59:04 -0400 From: Colin McLarty To: categories@mta.ca Date: Sun, 02 Mar 2008 19:04:13 -0500 MIME-Version: 1.0 Subject: categories: Re: Minimal abelian subcategory Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 4 > Now let R be a ring and M an R-module. Is there a minimal abelian > subcategory of Mod-R containing M? If so, is there a canonical way to > describe it? This question, as posed, is too easy: Just take M and its identity arrow. It will be a zero-object in that subcategory. There may be a better question here guiding Walter Mazorchuk's intuition, but it will have to require something more than just containing the one object. Colin From rrosebru@mta.ca Mon Mar 3 10:42:32 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 03 Mar 2008 10:42:32 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWBmo-0005g9-5m for categories-list@mta.ca; Mon, 03 Mar 2008 10:37:02 -0400 From: peasthope@shaw.ca Subject: categories: Re: A small cartesian closed concrete category To: categories@mta.ca MIME-version: 1.0 Content-type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Date: Mon, 03 Mar 2008 10:37:02 -0400 Status: O X-Status: X-Keywords: X-UID: 5 Folk, At Thu, 14 Feb 2008 15:06:49 -0500 I wrote, "Is there a cartesian closed concrete category which=20 is small enough to write out explicitly?" =20 At Fri, 15 Feb 2008 08:47:57 +0000 Philip Wadler srote, "... please summarize the replies ... and send ... to the ... list? ... interested to see if you receive a positive reply." I've counted 16 respondents! The question is=20 answered well. With my limited knowledge, the=20 summary probably fails to credit some of the=20 responses adequately but this is not intentional. Thanks to everyone who replied! 5 messages mentioned Hyting-algebras. Never heard of them. Lawvere & Schanuel=20 do not mention them in the 1997 book. =20 Will store the terms for future reference. Fred Linton wrote, "... skeletal version of the full category ... having as only objects the ordinal numbers 0 and 1. Here 0 x A =3D 0, 1 x A =3D A, 0^1 =3D 0, 0^0 =3D 1, 1^A =3D 1. In other words, B x A =3D min(A, B), B^A =3D max(1-A, B)." My product diagrams are at=20 http://carnot.yi.org/category01.jpg . Now I can try to illustrate the uniqueness=20 of map objects according to L&S, page page 314,=20 Exercise 1. Does this category have a name? =20 Is Boolean Category sensible? Two messages mentioned lambda calculus. Another topic for future reference. Stephen Lack asked "How small is small?=20 How explicit is explicit?" Probably=20 several other readers wondered the same. Fred's reply is small enough and explicit=20 enough to write out in detail. One message addressed the term "concrete". =20 I referred to Concrete Categories in the=20 Wikipedia. Matt Hellige mentioned categories a little=20 bigger than that described by Fred. =20 For instance, objects 0, 1, 2, 3. Map A -> B exists iff A < B. B x A =3D? min(A, B) =20 I should sketch the details of some of these=20 examples beyond the 0, 1 case above. Andrej Bauer described Fred's category in the context=20 of Heyting algebra and noted a proof by=20 Peter Freyd. Thorsten Altenkirch mentioned an equational=20 inconsistency which is beyond my present=20 grasp. Apologies to anyone who's reply is not =20 addressed adequately. If someone requests,=20 I can revise the summary and resubmit it. Thanks, ... Peter E. Desktops.OpenDoc http://carnot.yi.org/ From rrosebru@mta.ca Mon Mar 3 20:30:05 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 03 Mar 2008 20:30:05 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWKrd-00034P-GP for categories-list@mta.ca; Mon, 03 Mar 2008 20:18:37 -0400 Content-Disposition: inline Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset="utf-8" MIME-Version: 1.0 From: To: categories@mta.ca Subject: categories: Re: A small cartesian closed concrete category Date: Mon, 03 Mar 2008 16:30:55 -0500 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 6 Peter Easthope points out that in Lawvere & Schanuel there is no mention of Arend Heyting. That is unfortunate, especially since=20 pp 348-352 are devoted to introducing Heyting's Algebras=20 and one of their possible objective origins. The 2nd edition should correct this omission. Summarizing the 16 responses, a common thought of many must=20 have been=20 "If small implies finite then any example must be a poset (category in which any two parallel maps are equal) because of Freyd's theorem. A CC poset is almost=20 by definition a Heying Algebra. There are linearly ordered ones of=20 any size, but if the size is four or more, there are also examples that are not=20 linearly ordered.... =20 On the other hand if infinite examples=20 are allowed, and posetal ones are not, it is hard to think of a CCC smaller than a skeletal category of all finite sets." Bill From rrosebru@mta.ca Mon Mar 3 20:30:05 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 03 Mar 2008 20:30:05 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWKr2-00030x-8I for categories-list@mta.ca; Mon, 03 Mar 2008 20:18:00 -0400 MIME-version: 1.0 Content-transfer-encoding: 7BIT Content-type: TEXT/PLAIN; charset=US-ASCII; format=flowed Date: Mon, 03 Mar 2008 12:15:02 -0500 (EST) From: Joshua P Nichols-Barrer To: categories@mta.ca Subject: categories: Re: Minimal abelian subcategory Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 7 Hmm. I suppose that restricting to subcategories which respect the group structure on the Hom-sets would be enough to render the problem harder (the group structure of course can be recovered canonically from the underlying category, so this merely refines the class of subcategories we are considering). I would imagine this restriction would also have more repercussions for algebra, anyway... Josh On Sun, 2 Mar 2008, Colin McLarty wrote: >> Now let R be a ring and M an R-module. Is there a minimal abelian >> subcategory of Mod-R containing M? If so, is there a canonical way to >> describe it? > > This question, as posed, is too easy: Just take M and its identity > arrow. It will be a zero-object in that subcategory. There may be a > better question here guiding Walter Mazorchuk's intuition, but it will > have to require something more than just containing the one object. > > Colin > > > From rrosebru@mta.ca Tue Mar 4 08:49:19 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 04 Mar 2008 08:49:19 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWWVN-0004uo-2H for categories-list@mta.ca; Tue, 04 Mar 2008 08:44:25 -0400 Date: Mon, 03 Mar 2008 17:59:40 -0800 From: Vaughan Pratt MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Re: A small cartesian closed concrete category Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 8 > 5 messages mentioned Hyting-algebras. > Never heard of them.Lawvere & Schanuel > do not mention them in the 1997 book. > Will store the terms for future reference. Nowadays when I hear "Never heard of x" my subconscious seems to turn it into "never heard of Wikipedia." When five people tell you x is the answer to your question, merely filing it "for future reference" misses the point of the answer. (As one of the five, my examples consisted of the finite nonempty chains and the finite Boolean algebras, which I pointed out to Peter gave an example of every finite positive cardinality, and two for the powers of two. My mistake was to lump these examples together under the common rubric of "Heyting algebra," which appears to have made what was meant to be a simple answer incomprehensible.) As Bill points out, a Heyting algebra is almost the same thing as a CCC in the case of categories that are posets. This is exactly the case when there are finitely many objects (a case where Heyting algebras and distributive lattices are "the same thing" in the sense that they have the same underlying posets), and is close to true modulo existence of joins in the infinite case. In particular a Heyting algebra needs the empty join 0 in order to define negation as x->0, whence the negative integers made a category with its standard ordering is cartesian closed but is not a Heyting algebra for want of a least negative integer. More generally Heyting algebras are required to have all finite joins, not a requirement for posetal cartesian closed categories. Vaughan Pratt From rrosebru@mta.ca Tue Mar 4 08:49:19 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 04 Mar 2008 08:49:19 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWWTu-0004mA-Je for categories-list@mta.ca; Tue, 04 Mar 2008 08:42:54 -0400 Date: Tue, 4 Mar 2008 00:53:39 -0000 (GMT) Subject: categories: Minimal abelian subcategory (corrected) From: "Tom Leinster" To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 9 Apologies for the previous trivial question. Here is the correct version= . (The mistake was omitting to say that the subcategory must contain all endomorphisms of M.) * My colleague Walter Mazorchuk has the following question. Being abelian is a *property* of a category, not extra structure. Given an abelian category A, it therefore makes sense to define a subcategory o= f A to be an ABELIAN SUBCATEGORY if, considered as a category in its own right, it is abelian. Note that a priori, the inclusion need not preserv= e sums, kernels etc. Now let R be a ring and M an R-module. Is there a minimal abelian subcategory of Mod-R containing M and all its endomorphisms? If so, is there a canonical way to describe it? Any thoughts or pointers to the literature would be welcome. Feel free t= o assume hypotheses on R (it might be a finite-dimensional algebra etc), or to answer the question for full subcategories only. Thanks, Tom From rrosebru@mta.ca Tue Mar 4 22:21:11 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 04 Mar 2008 22:21:11 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWj1i-0006ee-4u for categories-list@mta.ca; Tue, 04 Mar 2008 22:06:38 -0400 Date: Tue, 04 Mar 2008 16:20:14 +0100 From: Andrej Bauer User-Agent: Thunderbird 2.0.0.12 (X11/20080227) MIME-Version: 1.0 To: Categories list Subject: categories: How to motivate a student of functional analysis Content-Type: text/plain; charset=ISO-8859-2; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 10 This semester I am teaching rudimentary category theory at graduate level. It is somewhat scary that I should be doing this, but other faculty members do not seem to do much general category theory. I have only few students (and they are very bright) but their areas of research are quite diverse: discrete math/computer science, algebra, algebraic topology, and functional analysis. I can plenty motivate categories for discrete math and computer science, with things like "initial algebras are inductive datatypes, final coalgebras are coinductive (lazy) datatypes". I also know enough general algebra to motivate algebraists with tquestions like "What is an additive category with a single object?". And we will study algebraic theories as well. Algebraic topologists are self-motivated. Nevertheless, we'll do some sheaves towards the end of the course. But how do I show the fun in categories to a student of functional analysis? I would like to give him a class project that he will find close to his interests. The course is covering (roughly) the following material: basic category theory (limits, colimits, adjoints, we mentioned additive and enriched categories), Lawvere's algebraic categories, monads (up to stating Beck's theorem and working out some examples), basics of presheaves and sheaves with a slant toward topology. There must be some functional analysis in there. I would very much appreciate some suggestions. Best regards, Andrej From rrosebru@mta.ca Tue Mar 4 22:21:11 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 04 Mar 2008 22:21:11 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWj35-0006mW-UU for categories-list@mta.ca; Tue, 04 Mar 2008 22:08:04 -0400 Date: Tue, 4 Mar 2008 23:49:39 +0000 (GMT) From: "Prof. Peter Johnstone" To: categories@mta.ca Subject: categories: Re: Minimal abelian subcategory (corrected) Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 11 On Tue, 4 Mar 2008, Tom Leinster wrote: > Apologies for the previous trivial question. Here is the correct version. > There's still something odd about this question. Requiring the subcategory to contain all endomorphisms of M of course requires it to contain A(M,M) as a monoid. But if you don't require it to be closed under biproducts in A, then presumably you don't require it to contain A(M,M) as a ring. It therefore raises two questions of "pure algebra": What conditions on a monoid (with 0) are needed to ensure that it occurs as the multiplicative monoid of a ring? Given that it does so occur, can there be several different additive group structures making it into a ring? I suspect that a fair amount must be known about these questions, but the only result I know in this area is one which I quoted in "Stone Spaces": for a ring of the form C(X), X a compact Hausdorff space, the multiplicative monoid structure of C(X) is enough to determine the topology of X (and hence the ring structure of C(X)) uniquely. Peter Johnstone > > My colleague Walter Mazorchuk has the following question. > > Being abelian is a *property* of a category, not extra structure. Given > an abelian category A, it therefore makes sense to define a subcategory of > A to be an ABELIAN SUBCATEGORY if, considered as a category in its own > right, it is abelian. Note that a priori, the inclusion need not preserve > sums, kernels etc. > > Now let R be a ring and M an R-module. Is there a minimal abelian > subcategory of Mod-R containing M and all its endomorphisms? If so, is > there a canonical way to describe it? > > Any thoughts or pointers to the literature would be welcome. Feel free to > assume hypotheses on R (it might be a finite-dimensional algebra etc), or > to answer the question for full subcategories only. > > Thanks, > Tom > > > > > From rrosebru@mta.ca Tue Mar 4 22:21:11 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 04 Mar 2008 22:21:11 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWj19-0006bt-Rt for categories-list@mta.ca; Tue, 04 Mar 2008 22:06:03 -0400 Mime-Version: 1.0 (Apple Message framework v624) Content-Type: text/plain; charset=US-ASCII; format=flowed Content-Transfer-Encoding: 7bit From: Paul Taylor Subject: categories: Heyting algebras and Wikipedia Date: Tue, 4 Mar 2008 14:17:18 +0000 To: Categories list Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 12 On the subject of Heyting algebras, usage seems to be ambiguous as to whether they should have (and their morphisms preserve) finite joins. I suggest that we should say "Heyting lattice" if they should, and "Heyting semilattice" if not. More generally, Vaughan said, > Nowadays when I hear "Never heard of x" my subconscious seems > to turn it into "never heard of Wikipedia." I too turn to Wikipedia for information on most subjects. For example its medical information is far superior to any other lay source that I have seen. But I have two reservations: Authority. Journalists like to take swipes at it on the grounds that anyone can edit it, but in my opinion they over-estimate the reliability of "authoritative" sources. A traditional paper encyclopedia consults only a small number of experts on each topic, so it's likely to be cliquey. On the other hand, there are frequently stories in www.TheRegister.co.uk (online geek news) about cliques taking over Wikipedia. Closer to home, the coverage of mathematics is extremely poor in comparison to other subjects. Usually, there is just the stark classical undergraduate definition, with neither advanced mainstream material nor any constructive critique. In my work on ASD, particularly its application to real analysis, I have wanted to refer to classical sources as a background, but on none of the relevant topics have I considered the Wikipedia article to be anywhere near satisfactory. All spaces are Hausdorff, and Excluded Middle is a Fact. I have thought about rewriting the articles on Dedekind cuts, locally compact spaces and some other things, but am afraid that my contributions will just be "reverted". Maybe if other categorists and constructivists joined in too, I would feel in better company. No, I don't want knock Wikipedia. It's a Good Thing, in principle. And I would like to encourage others to improve the mathematical coverage. By the way, there's also PlanetMath.org, in which authors "own" their articles, unless they have demonstrably abandoned them. Since I'm here, I would like to point out that there are thoroughly revised versions of The Dedekind Reals in ASD (with Andrej Bauer) and A Lambda Calculus for Real Analysis on my web page at www.PaulTaylor.EU/ASD/analysis.php The second of these contains a "need to know" introduction to the Scott topology, proof theory and the lambda calculus, ie it is written with the general mathematical audience in mind. Paul Taylor From rrosebru@mta.ca Wed Mar 5 09:25:19 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Mar 2008 09:25:19 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWtYp-0004ID-Lf for categories-list@mta.ca; Wed, 05 Mar 2008 09:21:31 -0400 Date: Wed, 5 Mar 2008 11:22:02 +0000 (GMT) From: "Prof. Peter Johnstone" To: categories@mta.ca Subject: categories: Re: Minimal abelian subcategory (corrected) MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 13 As George Janelidze pointed out to me, there was an error in what I wrote yesterday: the multiplicative monoid structure of C(X) determines X, and hence the ring structure of C(X), up to isomorphism (this is a 1949 result of A.N. Milgram), but it doesn't determine the additive structure uniquely, since one can take the standard addition and "conjugate" it by a multiplicative automorphism of R, in the same way that Steve points out for finite fields (e.g. one could define a new addition by f +' g = (f^3 + g^3)^{1/3}). Peter Johnstone On Wed, 5 Mar 2008, Steve Vickers wrote: > Dear Peter, > > A special case is that of groups (with 0 adjoined) and fields. But even that > is hard. In Paul Cohn's book on Skew Fields (p.145 of 1995 edition) he says > "The problem of characterizing the multiplicative group of a field has not > yet been solved even in the commutative case, but there are some results on > the subgroups of fields." (The main result he cites, one of Amitsur's, says > that a finite group G can be embedded as a subgroup in the multiplicative > group of a [skew]field if and only if G is (i) cyclic, or (ii) a certain kind > of metacyclic group, or (iii) a certain from of soluble goup with a > quaternion subgroup, or (iv) the binary icosahedral group SL_2(F_5) of order > 120.) > > Even in finite fields F = Z/p, the additive structure is not determined on > the nose by the multiplicative structure. If the group F* has a non-identity > automorphism alpha then (extending alpha to take 0 to 0) a different addition > for the same multiplication can be defined by x +' y = alpha^{-1}(alpha(x) + > alpha(y)). This would be the original addition only of alpha preserves > addition; but then since it preserves 0 and 1, and 1 is an additive > generator, then it would have to be the identity. An example is F_5, where > the multiplicative group is cyclic of order 4 and has a non-identity > automorphism that swaps the two generators. > > There remains the deeper question of whether you can have non-isomorphic > additive groups for the same multiplicative group. > > Regards, > > Steve. > > Prof. Peter Johnstone wrote: >> On Tue, 4 Mar 2008, Tom Leinster wrote: >> >> > Apologies for the previous trivial question. Here is the correct >> > version. >> > >> There's still something odd about this question. Requiring the subcategory >> to contain all endomorphisms of M of course requires it to contain >> A(M,M) as a monoid. But if you don't require it to be closed under >> biproducts in A, then presumably you don't require it to contain A(M,M) >> as a ring. It therefore raises two questions of "pure algebra": >> >> What conditions on a monoid (with 0) are needed to ensure that it occurs >> as the multiplicative monoid of a ring? >> >> Given that it does so occur, can there be several different additive >> group structures making it into a ring? >> >> I suspect that a fair amount must be known about these questions, but >> the only result I know in this area is one which I quoted in "Stone >> Spaces": for a ring of the form C(X), X a compact Hausdorff space, >> the multiplicative monoid structure of C(X) is enough to determine >> the topology of X (and hence the ring structure of C(X)) uniquely. >> >> Peter Johnstone > From rrosebru@mta.ca Wed Mar 5 09:25:19 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Mar 2008 09:25:19 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWtXO-00049g-Pu for categories-list@mta.ca; Wed, 05 Mar 2008 09:20:02 -0400 Date: Wed, 05 Mar 2008 01:11:41 -0800 From: Vaughan Pratt MIME-Version: 1.0 To: Categories list Subject: categories: Re: Heyting algebras and Wikipedia Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 14 Paul Taylor wrote: > In my work > on ASD, particularly its application to real analysis, I have > wanted to refer to classical sources as a background, but on none > of the relevant topics have I considered the Wikipedia article > to be anywhere near satisfactory. All spaces are Hausdorff, and > Excluded Middle is a Fact. I have thought about rewriting the > articles on Dedekind cuts, locally compact spaces and some other > things, but am afraid that my contributions will just be "reverted". In my understanding of Heyting algebras/lattices/semilattices, excluded middle fails for the algebras themselves but not for my understanding of them, where the partial order x <= y in a Heyting algebra is either true or false with no middle ground allowed. I have had little luck absorbing the logic of Heyting algebras into my own mathematical thinking. I furthermore worry that if ever I were to succeed my insights might become even less penetrating than they already are. On a related note, a careful reading of Max Kelly's "Basic Concepts of Enriched Category Theory" reveals that it is thoroughly grounded in Set, as I pointed out in August 2006 in my initial Wikipedia article on Max. I gave some thought to how one might eliminate Set from the treatment, without much success, and concluded that Max's judgment there was spot on. My feeling about these recommended Brouwerian modes of thoughts is that they are something like locker room accounts of social and other conquests: great stories about things that never actually happened, but which with sufficient repetition convince one that they must surely have occurred. The self-evident is merely an hypothesis that is so convenient, and that has been assumed for so long, that we can no longer imagine it false. This is just as true for Excluded Middle itself as for its negation. I happen to find Excluded Middle more convenient than its negation, but that's just me and perhaps others have had the opposite experience. Then there are those who accept neither Excluded Middle nor its negation, which takes us into the Hall of Mirrors that I always find myself in when I go down this particular rabbit-hole. Vaughan Pratt From rrosebru@mta.ca Wed Mar 5 09:25:19 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Mar 2008 09:25:19 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWtY6-0004E5-Cp for categories-list@mta.ca; Wed, 05 Mar 2008 09:20:46 -0400 Date: Wed, 05 Mar 2008 10:34:35 +0000 From: Steve Vickers MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Minimal abelian subcategory (corrected) Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 15 Dear Peter, A special case is that of groups (with 0 adjoined) and fields. But even that is hard. In Paul Cohn's book on Skew Fields (p.145 of 1995 edition) he says "The problem of characterizing the multiplicative group of a field has not yet been solved even in the commutative case, but there are some results on the subgroups of fields." (The main result he cites, one of Amitsur's, says that a finite group G can be embedded as a subgroup in the multiplicative group of a [skew]field if and only if G is (i) cyclic, or (ii) a certain kind of metacyclic group, or (iii) a certain from of soluble goup with a quaternion subgroup, or (iv) the binary icosahedral group SL_2(F_5) of order 120.) Even in finite fields F = Z/p, the additive structure is not determined on the nose by the multiplicative structure. If the group F* has a non-identity automorphism alpha then (extending alpha to take 0 to 0) a different addition for the same multiplication can be defined by x +' y = alpha^{-1}(alpha(x) + alpha(y)). This would be the original addition only of alpha preserves addition; but then since it preserves 0 and 1, and 1 is an additive generator, then it would have to be the identity. An example is F_5, where the multiplicative group is cyclic of order 4 and has a non-identity automorphism that swaps the two generators. There remains the deeper question of whether you can have non-isomorphic additive groups for the same multiplicative group. Regards, Steve. Prof. Peter Johnstone wrote: > On Tue, 4 Mar 2008, Tom Leinster wrote: > >> Apologies for the previous trivial question. Here is the correct >> version. >> > There's still something odd about this question. Requiring the subcategory > to contain all endomorphisms of M of course requires it to contain > A(M,M) as a monoid. But if you don't require it to be closed under > biproducts in A, then presumably you don't require it to contain A(M,M) > as a ring. It therefore raises two questions of "pure algebra": > > What conditions on a monoid (with 0) are needed to ensure that it occurs > as the multiplicative monoid of a ring? > > Given that it does so occur, can there be several different additive > group structures making it into a ring? > > I suspect that a fair amount must be known about these questions, but > the only result I know in this area is one which I quoted in "Stone > Spaces": for a ring of the form C(X), X a compact Hausdorff space, > the multiplicative monoid structure of C(X) is enough to determine > the topology of X (and hence the ring structure of C(X)) uniquely. > > Peter Johnstone From rrosebru@mta.ca Wed Mar 5 15:59:03 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Mar 2008 15:59:03 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWzjI-000606-9h for categories-list@mta.ca; Wed, 05 Mar 2008 15:56:44 -0400 From: "Katsov, Yefim" To: Date: Wed, 5 Mar 2008 10:25:36 -0500 Subject: categories: RE: How to motivate a student of functional analysis Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: quoted-printable MIME-Version: 1.0 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 16 Dear Adrei, May I suggest you to look at the monograph "Lectures and Exercises on Funct= ional Analysis" by A. Ya. Helemskii published by AMS in 2006, where, I'm su= re, you'll find a lot of good motivations for students interested in functi= onal analysis to study category theory. Good Luck and best regards, Yefim _______________________________________________________________________ Prof. Yefim Katsov Department of Mathematics & CS Hanover College Hanover, IN 47243-0890, USA telephones: office (812) 866-6119; home (812) 866-4312; fax (812) 866-7229 From rrosebru@mta.ca Wed Mar 5 15:59:03 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Mar 2008 15:59:03 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWzh4-0005er-TI for categories-list@mta.ca; Wed, 05 Mar 2008 15:54:26 -0400 From: Colin McLarty To: categories@mta.ca Date: Wed, 05 Mar 2008 09:24:02 -0500 MIME-Version: 1.0 Subject: categories: Re: Heyting algebras and Wikipedia Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 17 Vaughan Pratt Wednesday, March 5, 2008 8:32 am wrote, with much else: > On a related note, a careful reading of Max Kelly's "Basic Concepts of > Enriched Category Theory" reveals that it is thoroughly grounded in > Set,as I pointed out in August 2006 in my initial Wikipedia article > on Max. > I gave some thought to how one might eliminate Set from the > treatment,without much success, and concluded that Max's judgment > there was spot on. Without addressing this particular issue I want to say I appreciate the phrase in the article: "the explicitly foundational role of the category Set." I take it this is Vaughan's? Various people including Sol Feferman promote the view that if you use "sets" then you are admitting that you use ZF and not some categorical foundations. Vaughan's phrase goes aptly against that: If you use sets, then you use sets, but there is no reason it cannot be on categorical foundations. He does not say it *is* on categorical foundations, and that is fine in the context. He reminds people that it *could* be. best, Colin From rrosebru@mta.ca Wed Mar 5 15:59:03 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Mar 2008 15:59:03 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWzfs-0005Vf-Sg for categories-list@mta.ca; Wed, 05 Mar 2008 15:53:13 -0400 Date: Wed, 05 Mar 2008 16:10:43 +0100 From: Luigi Santocanale MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Postdoctoral research position in Theoretical Computer Science, Marseilles University Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 18 =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D POSTDOCTORAL RESEARCH POSITION IN THEORETICAL COMPUTER SCIENCE Marseilles University - CNRS - ANR CHOCO =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D The ANR project Curry-Howard for Concurrency (CHOCO) proposes a one year postdoc research position in Marseilles in the field of theoretical=20 computer science, starting in September 2008 (or as soon as possible=20 thereafter). The project CHOCO is focused on the applications of theoretical results=20 from mathematical logic and/or theoretical computer science to the=20 theory of concurrency. Candidates should have their PhD and a good background in at least one=20 of the following themes: - mathematical logic (lambda-calculus, complexity theory, linear logic), - semantics of programming languages (theory of categories,=20 denotationnal and game semantics), - models of concurrency (process calculi, bisimulation, event structures)= . The position will be taken in the logic group (LDP) of the Institut de Math=E9matiques de Luminy (IML); strong interaction is expected with the group MOVE of the Laboratoire d'Informatique Fondamentale (LIF) in=20 Marseilles, and the group Plume of the Laboratoire d'Informatique du=20 Parall=E9lisme in Lyon (LIP). Application should be sent to: postdoc-choco@choco.pps.jussieu.fr before May 18th 2008 and should include (all documents in pdf): - a CV (civil informations, universitary cursus, phd); - a work programme (no more than one page); - a publication list; - contact information for 2 references. Candidates will be notified by mid June. =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D CHOCO: http://choco.pps.jussieu.fr/ IML : http://iml.univ-mrs.fr/ LDP : http://iml.univ-mrs.fr/ldp/ LIF : http://www.lif.univ-mrs.fr/ MOVE : http://www.lif.univ-mrs.fr/spip.php?article89 LIP : http://www.ens-lyon.fr/LIP/web/ Plume: http://www.ens-lyon.fr/LIP/PLUME/index.html.en --=20 Luigi Santocanale LIF/CMI Marseille T=E9l: 04 91 11 35 74 http://www.cmi.univ-mrs.fr/~lsantoca/ Fax: 04 91 11 36 02 =09 From rrosebru@mta.ca Wed Mar 5 15:59:03 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Mar 2008 15:59:03 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWzf1-0005Nt-Qs for categories-list@mta.ca; Wed, 05 Mar 2008 15:52:19 -0400 Date: Wed, 5 Mar 2008 08:38:35 -0500 (EST) From: Michael Barr To: Categories list Subject: categories: Re: How to motivate a student of functional analysis MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 19 A student interested in functional analysis presumably knows some about topological vector spaces in general and Mackey spaces in particular. He might be interested in knowing that the full subcategory of Mackey spaces has a *-autonomous structure. This means that if M and N are Mackey there is a topology on the vector space of continuous linear maps M --> N that makes it into a Mackey space, often denoted M -o N, and that if you let M* = M -o C, then the canonical map M --> M** is an isomorphism. There is also a tensor product @ and the usual isomorphism Hom(M@N,P) = Hom(M,N-oP). See M. Barr, On $*$-autonomous categories of topological vector spaces. \cahiers {41} (2000), 243--254. From rrosebru@mta.ca Wed Mar 5 16:00:49 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Mar 2008 16:00:49 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWzmc-0006dN-Dq for categories-list@mta.ca; Wed, 05 Mar 2008 16:00:10 -0400 Content-Disposition: inline Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset="iso-8859-2" From: To: Categories list Subject: categories: Re: How to motivate a student of functional analysis Date: Wed, 05 Mar 2008 11:42:53 -0500 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 20 Functional Analysis was one of the key origins of categorical concepts and outlook, for example that the functionals themselves should be collected=20 into a single object (Voltera-Hadamard) leads to the Hom functor,etc. This was also one of the roads followed by students in the 1950s, for example from J. L. Kelley's "galactic" treatment of M. H. Stone's functor C. However in North America (as distinct from Europe) more recent functional analysists have accepted categorical methods only=20 grudgingly, and hence piecemeal. On the other hand, students who are not=20 specializing in analysis are often woefully ignorant of the basics of functional analysis that are part=20 of what every mathematician should know. To combat=20 that ignorance in my Advanced Graduate Algebra=20 course I often devoted several weeks to topics from functional analysis. It is a source of examples both interesting and essential. To begin to try to answer Andrej's question, I=20 rapidly recall some examples, and hope others will also comment: The double dual functor on Banch spaces is a protype example of a composite of adjoints becoming a monad. The EM algebras for this monoid were=20 computed by Fred Linton, in an exercise that should=20 be better known. It also illustrates the "descent" principle that C. Houzel cited couple of months ago (what I called semantics of structure of a given functor=20 in my thesis) : Objects constructed by a given functor tend to have, by virtue of that, more structure than=20 originally contemplated in its codomain , hence a lifted version of the functor comes closer to being invertible. As Peter Johnstone just recalled, if we consider commutative monoids with zero and hom them into the particular object of reals, the resulting set is "actually" a compact space whose C-algebra reveals by=20 adjointness that the opposite of the spaces form a full=20 subcategory of the monoids with zero. Again a good=20 exercise, related to Kelley's "square root lemma". Students might wonder why contiuous linear operators are traditionally called "bounded" (when they are not even). For many linear spaces (roughly those where sequentiality suffices) , preserving sequential limits is equivalent to preserving boundedness of sequences (for a linear map). George Mackey started to functorize this crucial observation before categories were fully explicit. Now we can consider the category of all presheaves on the category of all countable sets, define an "underlying" functor from Banach spaces to it, and verify that it actually lands in the subtopos of sheaves for the finite-disjoint-covering topology. Indeed it not only gives abelian group objects in the latter topos, but modules over R, the Dedekind reals of the topos, and a FULL subcategory of those. The above construction has an analogue using instead=20 Johnstone's coherent topos of sheaves on countable compact spaces. ETC Bill On Tue Mar 4 10:20 , Andrej Bauer sent: >This semester I am teaching rudimentary category theory at graduate >level. It is somewhat scary that I should be doing this, but other >faculty members do not seem to do much general category theory. > ... From rrosebru@mta.ca Wed Mar 5 16:01:59 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Mar 2008 16:01:59 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWznv-0006sB-RP for categories-list@mta.ca; Wed, 05 Mar 2008 16:01:31 -0400 Content-Disposition: inline Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset="us-ascii" MIME-Version: 1.0 From: To: categories@mta.ca Subject: categories: Re: Minimal abelian subcategory (corrected) Date: Wed, 05 Mar 2008 11:52:09 -0500 Message-Id: <6940.1204735929@buffalo.edu> Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 21 Oops Email is great. I cited Peter before he could correct himself.=20=20 A possible remedy would be to consider not just monoids with zero, but those equipped with a=20 homomorphism from R, so that points are retractions of that. This cuts down on the automorphisms, at least the naturally available ones. Bill On Wed Mar 5 6:22 , "Prof. Peter Johnstone" sent: >As George Janelidze pointed out to me, there was an error in what I wrote >yesterday: the multiplicative monoid structure of C(X) determines X, and >hence the ring structure of C(X), up to isomorphism (this is a 1949 >result of A.N. Milgram), but it doesn't determine the additive >structure uniquely, since one can take the standard addition and >"conjugate" it by a multiplicative automorphism of R, in the same way >that Steve points out for finite fields (e.g. one could define a new >addition by f +' g =3D (f^3 + g^3)^{1/3}). > >Peter Johnstone > From rrosebru@mta.ca Wed Mar 5 16:04:11 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Mar 2008 16:04:11 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWzpk-00078I-3R for categories-list@mta.ca; Wed, 05 Mar 2008 16:03:24 -0400 Date: Wed, 5 Mar 2008 17:31:00 -0000 (GMT) Subject: categories: Re: Minimal abelian subcategory (corrected) From: "Tom Leinster" To: categories@mta.ca MIME-Version: 1.0 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 22 A message from Walter Mazorchuk: Dear Colleagues, thank you very much for your comments on my abelian envelope question. Because of my stereotype thinking I missed the point of the determination of the additive structure by the multiplicative one in the original formulation. The stereotype is based on the fact that I am a representation theorist and the origin of the question is in module categories, which are k-linear over some field k. So, the subcategory I am looking for should be a k-linear subcategory with the induced k-linear structure. Best, Walter From rrosebru@mta.ca Wed Mar 5 16:07:15 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Mar 2008 16:07:15 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWzsZ-0007er-91 for categories-list@mta.ca; Wed, 05 Mar 2008 16:06:19 -0400 Mime-Version: 1.0 (Apple Message framework v753) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: Categories list Content-Transfer-Encoding: 7bit From: Thorsten Altenkirch Subject: categories: Re: Heyting algebras and Wikipedia Date: Wed, 5 Mar 2008 16:21:58 +0000 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 23 Hi Vaughan, On 5 Mar 2008, at 09:11, Vaughan Pratt wrote: > My feeling about these recommended Brouwerian modes of thoughts is > that > they are something like locker room accounts of social and other > conquests: great stories about things that never actually happened, > but > which with sufficient repetition convince one that they must surely > have > occurred. > > The self-evident is merely an hypothesis that is so convenient, and > that > has been assumed for so long, that we can no longer imagine it false. > This is just as true for Excluded Middle itself as for its > negation. I > happen to find Excluded Middle more convenient than its negation, but > that's just me and perhaps others have had the opposite experience. Indeed, being a computer scientist the BHK interpretation (see wikipedia) of logical connectives (which I can implement on a finite machine) makes more sense to me than the idea of infinite truth tables. You seem to think that the only alternative to excluded middle (forall P:Prop. P \/ not P) is (exists P:Prop. not (P \/ not P))? However, I'd say that "forall n:Nat. Halt n \/ not Halt n" is clearly invalid in the BHK interpretation without claiming that there is a particular statement which will never be decided, or a Turing machine which can be never shown to be terminating or not, i.e. even if we accepts Church's thesis, we arrive at "not (forall n:Nat. Halt n \/ not Halt n)" but not "exists n:Nat.not (Halt n \/ not Halt n)". To summarize, your reasoning seems to already presupposes that we accept Excluded Middle. > > Then there are those who accept neither Excluded Middle nor its > negation, which takes us into the Hall of Mirrors that I always find > myself in when I go down this particular rabbit-hole. > Maybe this is related to my reply? Thorsten This message has been checked for viruses but the contents of an attachment may still contain software viruses, which could damage your computer system: you are advised to perform your own checks. Email communications with the University of Nottingham may be monitored as permitted by UK legislation. From rrosebru@mta.ca Wed Mar 5 16:08:35 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Mar 2008 16:08:35 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWzuB-00008D-J0 for categories-list@mta.ca; Wed, 05 Mar 2008 16:07:59 -0400 From: "Ronnie" To: "Categories list" Subject: categories: Re: categories and Wikipedia Date: Wed, 5 Mar 2008 18:44:00 -0000 MIME-Version: 1.0 Content-Type: text/plain;format=flowed; charset="iso-8859-1";reply-type=response Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 24 I have made some minor contributions to wikipedia with information on for example John Robinson, on groupoids, Grothendieck, and the van Kampen theorem. The last three link to my web site and my counter (which registers `came from') shows the utility of these links, among many others. In the past a text had to assume or give an account of basic material. Why give an account of say Yoneda when there is a reasonable one on wiki which a reader can download? So I would encourage category theorists to develop the accounts. Ronnie ----- Original Message ----- From: "Paul Taylor" To: "Categories list" Sent: Tuesday, March 04, 2008 2:17 PM Subject: categories: Heyting algebras and Wikipedia > On the subject of Heyting algebras, usage seems to be ambiguous > as to whether they should have (and their morphisms preserve) > finite joins. I suggest that we should say "Heyting lattice" > if they should, and "Heyting semilattice" if not. > > More generally, Vaughan said, > > Nowadays when I hear "Never heard of x" my subconscious seems > > to turn it into "never heard of Wikipedia." > ... From rrosebru@mta.ca Wed Mar 5 16:11:04 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Mar 2008 16:11:04 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWzwc-0000Yz-A2 for categories-list@mta.ca; Wed, 05 Mar 2008 16:10:30 -0400 Date: Wed, 5 Mar 2008 14:30:43 -0500 (EST) From: Jeff Egger Subject: categories: Re: How to motivate a student of functional analysis To: Categories list MIME-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 25 Dear Andrej, Conventional functional analysis is largely concerned with Banach=20 spaces, and there's certainly alot that can be said about the=20 category of Banach spaces and linear contractions (i.e., continuous=20 linear transformations with norm less-than-or-equal-to 1). =20 For example, it is symmetric monoidal closed (with internal hom,=20 the space of _all_ continuous linear transformations!) and locally=20 countably presentable. In fact, it is a countable-ary quasi-variety,=20 and the full subcategory of its finite-dimensional objects is a good=20 example of a *-autonomous category that is not compact closed. =20 [Although it is not hard to prove the latter directly, it can also=20 be seen as an interesting application of Robin Houston's theorem=20 that products and coproducts can not differ in a compact closed=20 category.] =20 In fact, I think Ban is a fine example which can teach any student=20 of category theory a number of salutary lessons: 1. in category theory, the meaning of isomorphism is fixed---so if=20 you have a pre-existing class of isomorphisms in mind (in this case,=20 the isometric (norm-preserving) isomorphisms), then you must take=20 care in choosing an appropriate class of morphisms; 2a. there's more to defining internal homs than just slapping an =20 extra structure on the external homs;=20 2b. forgetful functors don't have to be "the obvious thing"; 3. you can't always have your cake and eat it too!---the whole=20 category can not hope to be self-dual, precisely because it is=20 locally presentable (and not a poset). Of course there are also (unital) C*-algebras, and I can make an=20 interesting point about them too---sometimes one needs to consider=20 maps between C*-algebras which are not *-homomorphisms: for example, there are "completely positive maps" and "completely bounded maps". Now, as important as the b.o./f.f. factorisation may be in general,=20 it seems fishy to speak of a category whose objects are C*-algebras but whose morphisms preserve only part of the C*-algebraic structure; and so it was that analysts were led to develop the notions of=20 "operator space" and "operator system" which provide the correct=20 level of structure to define c.b. maps and c.p. maps, respectively. =20 In fact, these are quite interesting categories in their own right: operator spaces are said to model "non-commutative functional analysis" ---but I only have a tenuous grasp of what that is supposed to mean! I meant to discuss quantale theory and Banach sheaves too, but I've run out of time---perhaps someone else will pick up the thread. Cheers, Jeff. --- Andrej Bauer wrote: > This semester I am teaching rudimentary category theory at graduate > level. It is somewhat scary that I should be doing this, but other > faculty members do not seem to do much general category theory. >=20 > I have only few students (and they are very bright) but their areas of > research are quite diverse: discrete math/computer science, algebra, > algebraic topology, and functional analysis. >=20 > I can plenty motivate categories for discrete math and computer science= , > with things like "initial algebras are inductive datatypes, final > coalgebras are coinductive (lazy) datatypes". >=20 > I also know enough general algebra to motivate algebraists with > tquestions like "What is an additive category with a single object?". > And we will study algebraic theories as well. >=20 > Algebraic topologists are self-motivated. Nevertheless, we'll do some > sheaves towards the end of the course. >=20 > But how do I show the fun in categories to a student of functional > analysis? I would like to give him a class project that he will find > close to his interests. The course is covering (roughly) the following > material: basic category theory (limits, colimits, adjoints, we > mentioned additive and enriched categories), Lawvere's algebraic > categories, monads (up to stating Beck's theorem and working out some > examples), basics of presheaves and sheaves with a slant toward > topology. There must be some functional analysis in there. >=20 > I would very much appreciate some suggestions. >=20 > Best regards, >=20 > Andrej >=20 >=20 >=20 Looking for the perfect gift? Give the gift of Flickr!=20 http://www.flickr.com/gift/ From rrosebru@mta.ca Wed Mar 5 16:12:36 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Mar 2008 16:12:36 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JWzy4-0000qe-OQ for categories-list@mta.ca; Wed, 05 Mar 2008 16:12:01 -0400 Date: Wed, 05 Mar 2008 10:13:55 -0800 From: PETER EASTHOPE Subject: categories: Re^3: A small cartesian closed concrete category To: categories@mta.ca MIME-version: 1.0 Content-type: text/plain; charset=us-ascii Content-language: en Content-transfer-encoding: 7bit Content-disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 26 Vaughan P., vp> ... "never heard of Wikipedia." When five people tell you x is the answer to your question, ... I appreciate your exasperation. Am afraid that almost everyone who replied to my question is severely over-estimating my state of comprehension. My background is primarily in engineering and physics, whereas most of you teach at the honours undergraduate and graduate levels. At present I am trying to understand the concept of map object and the exercises on pp. 314 and 315 of L&S. Long ago, a professional mathematician, as you all are, advised: When stuck, find examples until you come to an understanding. Presently I am looking for examples illustrating Exercises 1-6. Fred's reply is a good start. Longer chains suggested by Matt H. will be interesting, if not necessary. Skipping ahead 34 pages, I see that map objects are an ingredient of a topos. A scan of http://en.wikipedia.org/wiki/Heyting_algebra reports that "map object" is not in the page. Perhaps it should be. Neverthless, working through the book systematically seems more promising than reading about toposes and Heyting algebras before understanding map objects. fwl> ... Heyting's Algebras and one of their possible objective origins. The 2nd edition should correct this omission. I don't want to be presumptuous, but if some of the tiny categories mentioned by Fred and Matt can also fit into the second edition, that would certainly interest me. Without this text, my endeavour to learn category th. would be quite a battle. Thanks! I should have explained at the beginning, the intention in seeking the examples. Sorry for the aggravation. Regards, ... Peter E. http://carnot.yi.org/ From rrosebru@mta.ca Thu Mar 6 09:42:38 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 06 Mar 2008 09:42:38 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXGD3-0006Rr-99 for categories-list@mta.ca; Thu, 06 Mar 2008 09:32:33 -0400 Date: Wed, 05 Mar 2008 20:22:37 -0500 From: "Fred E.J. Linton" To: Categories list Subject: categories: Re: How to motivate a student of functional analysis Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 27 Following Jeff Egger, who wrote, in part, "Ban is a fine example which can teach any student of category theory a number of salutary lessons," but asking forgiveness for tooting my own horn, I'd like to point out = another one of those lessons -- my old characterization of Banach = conjugate spaces as the algebras over the double-dualization monad = on {Ban}. Neat mix of Beck Theorem, functional analysis, and more, on pp. 227-240 of: = Proc. Conf. Integration, Topology, and Geometry in Linear Spaces, in: Contemporary Mathematics, Volume 2, AMS, Providence, 1980. Might even serve as one student's "individual reading report" project. There's also my even older squib on "Functorial Measure Theory," in pp. 36-49 of: Proc. Conf. Functional Analysis, UC Irvine, 3/28-4/1, 1966, Thompson Book Co., Wash., DC, & Academic Press, London, 1967. This one breathes life into the slogan, "Measures are adjoint to functions." Cheers, -- Fred From rrosebru@mta.ca Thu Mar 6 09:42:39 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 06 Mar 2008 09:42:39 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXGDt-0006Yt-MM for categories-list@mta.ca; Thu, 06 Mar 2008 09:33:25 -0400 From: Robert L Knighten MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Date: Wed, 5 Mar 2008 18:37:11 -0800 To: Categories list Subject: categories: How to motivate a student of functional analysis Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 28 Most of the material connecting analysis and category theory seems to be written by specialists in category theory who have observed some of the ways that insights from category theory can be brought to bear (for example look on Math Sci Net at the many papers (co)authored by Joan Wick Pelletier.) But here's an example which is a book on functional analysis that has a strong use of categories: @book {MR0296671, AUTHOR = {Semadeni, Zbigniew}, TITLE = {Banach spaces of continuous functions. {V}ol. {I}}, NOTE = {Monografie Matematyczne, Tom 55}, PUBLISHER = {PWN---Polish Scientific Publishers}, ADDRESS = {Warsaw}, YEAR = {1971}, PAGES = {584 pp. (errata insert)}, MRCLASS = {46E15 (46M99)}, MRNUMBER = {MR0296671 (45 \#5730)}, MRREVIEWER = {H. E. Lacey}, } -- Bob -- Robert L. Knighten RLK@knighten.org From rrosebru@mta.ca Thu Mar 6 09:42:39 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 06 Mar 2008 09:42:39 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXGCU-0006Li-Ro for categories-list@mta.ca; Thu, 06 Mar 2008 09:31:58 -0400 Date: Wed, 5 Mar 2008 17:53:54 -0500 (EST) From: Michael Barr To: Categories list Subject: categories: graphics and dvi MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 29 If you read the TAC instructions for authors, you will find that we discourage the use of the graphics package unless absolutely necessary because, for the time being, dvi is still our basic archive format and graphics specials were not rendered properly by dvi viewers. I recently discovered that at least one dvi viewer does indeed render graphics specials. Namely the yap viewer that comes with miktex, a windows-specific implementation of tex enters something called dvips mode (meaning, I imagine, an on-the fly conversion to ps and then rendering that). To be more precise, miktex2.5 asks if you want dvips mode (why wouldn't you?) and miktex2.7 enters it automatically. (I don't know about 2.6). As far as I know no Unix (or Linux) viewer does this. Neither pdflatex nor dvipdfm renders graphics specials correctly. The only way I have been able to get correct pdf files is to first use dvips to make a ps file and then use ghostscript to convert to pdf. Still, one can hope that these problems will disappear in some future implementations and we may withdraw our objections to the use of the graphics package. At that point, our reluctance to use Paul Taylor's diagrams will also disappear. For the time being, however, we will continue to recommend the use of xy-pic and, in particular, the diagxy front end. From rrosebru@mta.ca Thu Mar 6 09:42:39 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 06 Mar 2008 09:42:39 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXGHP-00079l-Ce for categories-list@mta.ca; Thu, 06 Mar 2008 09:37:03 -0400 Date: Thu, 06 Mar 2008 11:15:57 +0000 To: categories@mta.ca Subject: categories: CiE 2008 - accepted papers, informal presentations, participation MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit From: A.Beckmann@swansea.ac.uk (Arnold Beckmann) Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 30 [Apologies for multiple copies] **************************************************************** Computability in Europe 2008: Logic and Theory of Algorithms University of Athens, June 15-20 2008 http://www.cs.swan.ac.uk/cie08/ CONTENTS: 1) List of accepted papers 2) Call for informal presentations 3) Call for participation 1) Accepted papers The list of accepted papers can be found at http://www.cs.swan.ac.uk/cie08/give-page.php?18 2) Informal Presentations There is a remarkable difference in conference style between computer science and mathematics conferences. Mathematics conferences allow for informal presentations that are prepared very shortly before the conference and inform the participants about current research and work in progress. The format of computer science conferences with pre-produced proceedings volumes is not able to accommodate this form of scientific communication. Continuing the tradition of past CiE conferences, also this year's CiE conference endeavours to get the best of both worlds. In addition to the formal presentations based on our LNCS proceedings volume, we invite researchers to present informal presentations. For this, please send us a brief description of your talk (between one paragraph and half a page) before 30 April 2008. Please submit your abstract via our Submission Form, now online at: http://www.cs.swansea.ac.uk/cie08/abstract-submission.php You will be notified whether your informal presentation has been accepted before 15 May 2008. Let us remind you that there will be three post-conference publications of CiE 2008, see http://www.cs.swansea.ac.uk/cie08/publications.php All speakers, including the speakers of informal presentations, are eligible to be invited to submit a full journal version of their talk to one of the post-conference publications. 3) Registration for CiE 2008 is now open: http://www.cs.swan.ac.uk/cie08/registration.php The early registration deadline is 4 May 2008. You can also use the registration process to book accommodation. Please note that the current prices as listed on our website http://www.cs.swan.ac.uk/cie08/accommodation.php are only guaranteed until 31 March 2008. From rrosebru@mta.ca Thu Mar 6 09:42:40 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 06 Mar 2008 09:42:40 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXGHp-0007D9-JG for categories-list@mta.ca; Thu, 06 Mar 2008 09:37:29 -0400 Date: Thu, 06 Mar 2008 12:12:31 +0000 To: categories@mta.ca Subject: categories: CiE 2008 - grants MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit From: A.Beckmann@swansea.ac.uk (Arnold Beckmann) Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 31 [Apologies for multiple copies] **************************************************************** Computability in Europe 2008: Logic and Theory of Algorithms University of Athens, June 15-20 2008 http://www.cs.swan.ac.uk/cie08/ Call for Grant Applications Deadline: 15 APRIL, 2008 A number of grants are available for attenting CiE 2008. They are intended for students, post-docs and persons with limited means. Also, student members of the ASL may apply for travel funds. For more details see our website http://www.cs.swansea.ac.uk/cie08/grants.php From rrosebru@mta.ca Thu Mar 6 09:42:40 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 06 Mar 2008 09:42:40 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXGGF-0006zO-Ks for categories-list@mta.ca; Thu, 06 Mar 2008 09:35:51 -0400 Date: Wed, 05 Mar 2008 21:15:46 -0800 From: Vaughan Pratt MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Heyting algebras and Wikipedia Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 32 Colin is exactly right on all points. I tend to look at sets from the perspective neither of a set theorist nor a category theorist but a combinatorialist. As long as people agree on the cardinalities of the homsets between sets, particularly the finite ones, I figure they must be talking about essentially the same objects. Infinite domains are problematic for everyone, infinite codomains much less so (we understand the homset N^2 much better than 2^N). The remark in my post about the self-evident being merely a convenient long-held hypothesis (which I put on my "sayings" website http://boole.stanford.edu/dotsigs.html less than a month ago) applies in spades to membership as characteristic of sets, the premise for ZF. Those who identify acceptance of the category Set with acceptance of ZF have not only not accepted but not even grasped that the wholesale replacement of the binary relation of membership by the (partial) binary operation of composition, with a set of axioms radically different from those of ZF, is a foundational move. ZF is so deeply ingrained in their thought processes that they have no idea how to think about mathematical structures without falling back on its axioms. Borrowing from Hilbert, they are unable to replace "set," "function," and "composite" by "table," "chair," and "beermug." If you find it hard to imagine how anyone could find it hard to imagine mathematics without ZF, just read Steve Simpson on 2/25/98 (almost exactly a decade ago) at http://cs.nyu.edu/pipermail/fom/1998-February/001228.html The bit "I totally repudiate every syllable of every word of every subclaim of every claim that McLarty has ever made about what he is pleased to call `categorical foundations'" made abundantly clear back then that Steve could not begin to concieve of replacing membership by composition as the basis for an alternative foundation of mathematics. While I can't speak for Steve today, this remains a stumbling block for those raised to believe that rigorous mathematics would not be possible in a world where propositions such as "for all x and y there exists z such that x is a subset of z and y is a member of z" did not hold. How could x U {y} fail to exist and the walls of mathematics not come tumbling down? Vaughan Colin McLarty wrote: > Vaughan Pratt > Wednesday, March 5, 2008 8:32 am > > wrote, with much else: > >> On a related note, a careful reading of Max Kelly's "Basic Concepts of >> Enriched Category Theory" reveals that it is thoroughly grounded in >> Set,as I pointed out in August 2006 in my initial Wikipedia article >> on Max. >> I gave some thought to how one might eliminate Set from the >> treatment,without much success, and concluded that Max's judgment >> there was spot on. > > Without addressing this particular issue I want to say I appreciate the > phrase in the article: "the explicitly foundational role of the category > Set." I take it this is Vaughan's? > > Various people including Sol Feferman promote the view that if you use > "sets" then you are admitting that you use ZF and not some categorical > foundations. Vaughan's phrase goes aptly against that: If you use > sets, then you use sets, but there is no reason it cannot be on > categorical foundations. He does not say it *is* on categorical > foundations, and that is fine in the context. He reminds people that it > *could* be. > > best, Colin > > From rrosebru@mta.ca Thu Mar 6 09:42:40 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 06 Mar 2008 09:42:40 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXGGw-00075z-1c for categories-list@mta.ca; Thu, 06 Mar 2008 09:36:34 -0400 Date: Thu, 06 Mar 2008 11:35:59 +0100 From: Luigi Santocanale MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Call for participation: workshop on MODAL FIXPOINT LOGICS, Amsterdam, March 25-27 2008 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 33 [Apologies for multiple copies] Call for participation. Workshop on MODAL FIXPOINT LOGICS Amsterdam, March 25-27 2008 http://staff.science.uva.nl/~yde/mfl Registration deadline: March 19, 2008 Modal fixpoint logics constitute a research field of considerable interest, not only because of their many applications, but also because of their rich logical/mathematical theory. Systems such as LTL, PDL, CTL, and the modal mu-calculus, originate from computer science, and are for instance applied in the theory of program specification and verification. The richness of their theory stems from the deep connections with various fields in logic, mathematics, and theoretical computer science, such as lattices and universal (co-)algebra, modal logic, automata, and game theory. Large areas of the theory of modal fixpoint logics, in particular the connection with the theory of automata and games, have been intensively investigated and are by now are well understood. Nevertheless, there are still many aspects that are less explored. This applies in particular to the model theory, intended as the study of a logic as a function of classes of models, the proof theory, the algebraic logic, duality theory in the spirit of Stone/Priestly duality, and the relation to the theory of ordered sets as grounding the concept of "least fixpoint". The aim of the workshop is to bring together researchers from various backgrounds, in particular, computer scientists and pure logicians, who share an interest in the area. The workshop program is available from the web site=20 http://staff.science.uva.nl/~yde/mfl. Invited speakers: Marcello Bonsangue, Leiden Johan van Benthem, Amsterdam Dietmar Berwanger, Aachen Giovanna D'Agostino, Udine Dexter Kozen, Cornell Giacomo Lenzi, Pisa Damian Niwinski, Warszawa Colin Stirling, Edinburgh Thomas Studer, Bern Albert Visser, Utrecht Igor Walukiewicz, Bordeaux Thomas Wilke, Kiel Organizers: Luigi Santocanale, Marseille Yde Venema, Amsterdam --=20 Luigi Santocanale LIF/CMI Marseille T=E9l: 04 91 11 35 74 http://www.cmi.univ-mrs.fr/~lsantoca/ Fax: 04 91 11 36 02 =09 From rrosebru@mta.ca Thu Mar 6 09:42:41 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 06 Mar 2008 09:42:41 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXG9a-0005tj-2q for categories-list@mta.ca; Thu, 06 Mar 2008 09:28:58 -0400 Date: Wed, 05 Mar 2008 20:40:13 +0000 From: Tim Porter To: Categories list Subject: categories: Re: How to motivate a student of functional analysis MIME-Version: 1.0 Content-Type: text/plain;charset=ISO-8859-1;DelSp="Yes";format="flowed" Content-Disposition: inline Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 34 Jeff did not mention the excellent very categorical lectures given on=20=20= =0A= operator spaces by Matthias Neufang at the=0A= FIELDS INSTITUTE=0A= Summer School in Operator Algebras=0A= held last summer at the University of Ottawa and that we both=20=20=0A= attended. I do not know if any version of Matthias' notes is=20=20=0A= available. The theme of tensor products was important. Not only did=20=20= =0A= his lectures provide good motivation for studying the subject from a=20=20= =0A= categorical viewpoint. He did not do the category theory of operator=20=20= =0A= spaces but rather was explicitly conscious of the categorical content=20=20= =0A= of what he was saying. His notes may be of some interest to others so=20= =20=0A= let us hope he will put some of the material on the web.=0A= =0A= =0A= Tim=0A= =0A= =0A= =0A= =0A= Quoting Jeff Egger :=0A= =0A= > Dear Andrej,=0A= >=0A= > Conventional functional analysis is largely concerned with Banach=0A= > spaces, and there's certainly alot that can be said about the=0A= > category of Banach spaces and linear contractions (i.e., continuous=0A= > linear transformations with norm less-than-or-equal-to 1).=0A= >=0A= From rrosebru@mta.ca Thu Mar 6 14:03:16 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 06 Mar 2008 14:03:16 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXKOl-0007Ut-DS for categories-list@mta.ca; Thu, 06 Mar 2008 14:00:55 -0400 From: "Ronnie" To: "Categories list" Subject: categories: Re: How to motivate a student of functional analysis Date: Thu, 6 Mar 2008 15:15:11 -0000 MIME-Version: 1.0 Content-Type: text/plain;format=flowed; charset="iso-8859-1";reply-type=original Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 35 You could also look at MR1471480 (98i:58015) Kriegl, Andreas; Michor, Peter W. The convenient setting of global analysis. (English summary) Mathematical Surveys and Monographs, 53. American Mathematical Society, Providence, RI, 1997. x+618 pp. ISBN: 0-8218-0780-3 for which an e-version has been downloadable. However as the review says: "the exposition is based on functional analysis rather than on category theory; this fact will, undoubtedly, allow the subject to reach a wider audience. " Ronnie ----- Original Message ----- From: "Robert L Knighten" To: "Categories list" Sent: Thursday, March 06, 2008 2:37 AM Subject: categories: How to motivate a student of functional analysis > Most of the material connecting analysis and category theory seems to be > written by specialists in category theory who have observed some of the > ways > that insights from category theory can be brought to bear (for example > look on > Math Sci Net at the many papers (co)authored by Joan Wick Pelletier.) But > here's an example which is a book on functional analysis that has a strong > use > of categories: > > > @book {MR0296671, > AUTHOR = {Semadeni, Zbigniew}, > TITLE = {Banach spaces of continuous functions. {V}ol. {I}}, > NOTE = {Monografie Matematyczne, Tom 55}, > PUBLISHER = {PWN---Polish Scientific Publishers}, > ADDRESS = {Warsaw}, > YEAR = {1971}, > PAGES = {584 pp. (errata insert)}, > MRCLASS = {46E15 (46M99)}, > MRNUMBER = {MR0296671 (45 \#5730)}, > MRREVIEWER = {H. E. Lacey}, > } > > -- Bob > > -- > Robert L. Knighten > RLK@knighten.org > > > > > -- > No virus found in this incoming message. > Checked by AVG Free Edition. > Version: 7.5.516 / Virus Database: 269.21.4/1312 - Release Date: > 04/03/2008 21:46 > From rrosebru@mta.ca Thu Mar 6 14:03:16 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 06 Mar 2008 14:03:16 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXKQS-0007nE-M5 for categories-list@mta.ca; Thu, 06 Mar 2008 14:02:40 -0400 Date: Thu, 6 Mar 2008 12:35:44 -0500 (EST) From: Jeff Egger Subject: categories: Re: How to motivate a student of functional analysis To: Categories list MIME-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 36 Hi Tim, > Jeff did not mention the excellent very categorical lectures given on =20 > operator spaces by Matthias Neufang at the > FIELDS INSTITUTE > Summer School in Operator Algebras > held last summer at the University of Ottawa and that we both =20 > attended.=20 There are, of course, many people I could have credited and cited in=20 my previous posting, but only the newest readers of this list will be unaware of the perils with which such an attempt is fraught. [For=20 instance, I first read of the local presentability of Ban in Adamek=20 and Rosicky's book, but I would not like to hazard a guess as to the=20 origin of this result.] But you are right: I should have made an exception in Matthias' case. I should also credit Vladimir Pestov, a topologist who knows enough category theory to wonder whether operator spaces might be internal=20 Banach spaces in some Grothendieck topos (but perhaps not enough to=20 realise that this might take a student more than one term to prove), for having introduced me to Matthias several years ago. =20 While at Dalhousie, I gave a talk on Pestov's conjecture; but when=20 I started writing up my notes, I was distracted by an unrelated=20 observation about the category of operator spaces which ultimately=20 led to my ill-fated C*-algebra paper. I still haven't gotten back=20 to the original project. =20 > I do not know if any version of Matthias' notes is available.=20 Nor do I, but I am sure he would rather point people towards the=20 pre-Wikipedia-era "online dictionary" of operator space theory to=20 which he contributed:=20 [German] http://www.math.uni-sb.de/ag/wittstock/projekt99.html [English] http://www.math.uni-sb.de/ag/wittstock/projekt2001.html These notes are quite good in the sense that, to use Tim's words,=20 they are > explicitly conscious of the categorical content =20 In particular, it is quite gratifying to see a theorem such as "the=20 forgetful functor from operator spaces to Banach space admits both=20 a left and a right adjoint" stated (more or less) ungrudgingly. =20 Cheers, Jeff. P.S. I should say that I don't think that operator spaces would be a=20 suitable topic for an introductory CT course to a general audience;=20 they are rather intricate. But it might be possible to craft an=20 interesting set of exercises for the functional analysis contingent=20 of such a course around operator space theory. =20 From rrosebru@mta.ca Thu Mar 6 14:03:17 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 06 Mar 2008 14:03:17 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXKML-00079j-TZ for categories-list@mta.ca; Thu, 06 Mar 2008 13:58:25 -0400 From: Colin McLarty To: categories@mta.ca Date: Thu, 06 Mar 2008 09:10:52 -0500 MIME-Version: 1.0 Subject: categories: How to motivate me to become a student of functional analysis Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 37 Robert L Knighten Thursday, March 6, 2008 8:48 am mentioned Semadeni _Banach spaces of continuous functions_ as using a categorical perspective. Maybe that is the book I need. I want to understand Grothendieck's functional analysis in more detail than just to say he used categorical definitions of different tensor products to explain Fredholm kernels. For a start, I know nothing about Fredholm kernels except what is on Wikipedia. Grothendieck's own writings on it are long and start with many definitions so that it is hard for me to see the point -- he even says in Recoltes et Semailles that he never really *felt* the point but did it as an assignment. So that work shows nothing like the very clear motivation he gives for schemes and etale cohomology in SGA. What is a good introduction to his contributions in functional analysis? best, Colin From rrosebru@mta.ca Thu Mar 6 14:03:17 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 06 Mar 2008 14:03:17 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXKNi-0007JG-FL for categories-list@mta.ca; Thu, 06 Mar 2008 13:59:50 -0400 From: Pedro Resende Subject: categories: Re: How to motivate a student of functional analysis Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit Mime-Version: 1.0 (Apple Message framework v915) Date: Thu, 6 Mar 2008 14:19:15 +0000 To: Categories list Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 38 Every student who learned the basics of operator algebras knows the Gelfand-Naimark representation theorem, usually stated non- categorically as "every commutative unital C*-algebra is isomorphic to the algebra of continuous functions on a compact Hausdorff space". Asking such students to check that this is part of a dual equivalence of categories is probably a good idea, and based on this one can do exercises about particular algebras they know - for instance to compute presentations by generators and relations of C(T^n), the algebra of continuous functions on the n-dimensional torus, which by the duality is instantly reduced to finding a presentation of C(S^1), etc. Also, some students might be willing to work out how the existence of presentations of C*-algebras by generators and relations relates to the fact that the category of C*-algebras is algebraic over Sets. On Mar 4, 2008, at 3:20 PM, Andrej Bauer wrote: > This semester I am teaching rudimentary category theory at graduate > level. It is somewhat scary that I should be doing this, but other > faculty members do not seem to do much general category theory. > > I have only few students (and they are very bright) but their areas of > research are quite diverse: discrete math/computer science, algebra, > algebraic topology, and functional analysis. > > I can plenty motivate categories for discrete math and computer > science, > with things like "initial algebras are inductive datatypes, final > coalgebras are coinductive (lazy) datatypes". > > I also know enough general algebra to motivate algebraists with > tquestions like "What is an additive category with a single object?". > And we will study algebraic theories as well. > > Algebraic topologists are self-motivated. Nevertheless, we'll do some > sheaves towards the end of the course. > > But how do I show the fun in categories to a student of functional > analysis? I would like to give him a class project that he will find > close to his interests. The course is covering (roughly) the following > material: basic category theory (limits, colimits, adjoints, we > mentioned additive and enriched categories), Lawvere's algebraic > categories, monads (up to stating Beck's theorem and working out some > examples), basics of presheaves and sheaves with a slant toward > topology. There must be some functional analysis in there. > > I would very much appreciate some suggestions. > > Best regards, > > Andrej > > From rrosebru@mta.ca Thu Mar 6 14:07:08 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 06 Mar 2008 14:07:08 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXKUW-0001B4-OE for categories-list@mta.ca; Thu, 06 Mar 2008 14:06:52 -0400 MIME-version: 1.0 Content-transfer-encoding: 7BIT Content-type: text/plain; charset=us-ascii From: Dan Christensen To: Categories list Subject: categories: Re: graphics and dvi Date: Thu, 06 Mar 2008 12:41:30 -0500 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 39 [Note from moderator: Discussion of TeX-nicalities is admittedly off-topic, but the last paragraph below is of wide interest.] Michael Barr writes: > If you read the TAC instructions for authors, you will find that we > discourage the use of the graphics package unless absolutely necessary > because, for the time being, dvi is still our basic archive format and > graphics specials were not rendered properly by dvi viewers. I'm curious what you mean by this. One of my duties as Managing Editor of Homology, Homotopy and Applications is to oversee the copyediting and typesetting, and I produce the final versions of all files in dvi, ps and pdf format on my linux machine. We receive files using quite a variety of graphics packages, including the "graphics" package, and rarely have any problems with the resulting dvi files. (I test using xdvi.) The only problem I can recall is with figures that are rotated 90 degrees, and for such papers we simply don't make the dvi file publicly available. > Neither pdflatex nor dvipdfm renders graphics specials correctly. The pdf files for HHA are almost always produced using dvipdfm, and again this works quite reliably in my experience. In some cases, we use pdflatex, and again I have had no trouble with it. dvipdfm works even in the one or two cases where xdvi didn't display a file correctly, such as with rotated figures. And while for most files I regard dvi as the primary processed archive, I also use the snapshot package to save most of the .sty files each article includes, so that if necessary in the future the articles can be reprocessed to produce some hypothetical new format that contains information not in the dvi file. (E.g. to add hyperlinks to all articles.) While I am writing, I'd like to encourage readers of this list to submit articles to HHA and to ask their libraries to subscribe if they don't already. The price is quite reasonable, and we regularly publish articles with a categorical bent. All articles are available online at http://intlpress.com/HHA You can receive announcements of new articles by writing directly to me and asking to be put on the announcements mailing list. Best wishes, Dan From rrosebru@mta.ca Thu Mar 6 22:43:59 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 06 Mar 2008 22:43:59 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXSRF-0003FG-Jr for categories-list@mta.ca; Thu, 06 Mar 2008 22:36:01 -0400 Date: Thu, 6 Mar 2008 16:01:33 +0000 (GMT) From: Paul B Levy To: categories@mta.ca Subject: categories: Re: Re: Heyting algebras and Wikipedia MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 40 > While I can't speak for Steve today, this remains a stumbling block for > those raised to believe that rigorous mathematics would not be possible > in a world where propositions such as "for all x and y there exists z > such that x is a subset of z and y is a member of z" did not hold. How > could x U {y} fail to exist and the walls of mathematics not come > tumbling down? Maybe not the walls of mathematics, but what about theorems like "every polynomial functor on Set has a unique initial algebra whose structure map is an identity"? I think theorems like this are worth retaining (and antifoundation makes even more of them). I'd also like to suggest that "foundations" is being used in two very different senses. In FoM, it's about quantifying the philosophical risks involved in particular formal systems and proofs, i.e. issues such as relative consistency, omega-consistency, etc. For this purpose the primacy of membership vs composition is quite immaterial. One could, I suppose, make a formal theory based on composition equal in strength (in whatever sense) to ZF. Category theory on the other hand is about fundamental algebraic structures. I don't think it makes sense to ask "is category theory omega-consistent?" as one can for ZF (not that anyone knows the answer). Paul > > Vaughan > > Colin McLarty wrote: >> Vaughan Pratt >> Wednesday, March 5, 2008 8:32 am >> >> wrote, with much else: >> >>> On a related note, a careful reading of Max Kelly's "Basic Concepts of >>> Enriched Category Theory" reveals that it is thoroughly grounded in >>> Set,as I pointed out in August 2006 in my initial Wikipedia article >>> on Max. >>> I gave some thought to how one might eliminate Set from the >>> treatment,without much success, and concluded that Max's judgment >>> there was spot on. >> >> Without addressing this particular issue I want to say I appreciate the >> phrase in the article: "the explicitly foundational role of the category >> Set." I take it this is Vaughan's? >> >> Various people including Sol Feferman promote the view that if you use >> "sets" then you are admitting that you use ZF and not some categorical >> foundations. Vaughan's phrase goes aptly against that: If you use >> sets, then you use sets, but there is no reason it cannot be on >> categorical foundations. He does not say it *is* on categorical >> foundations, and that is fine in the context. He reminds people that it >> *could* be. >> >> best, Colin >> >> > > > -- Paul Blain Levy email: pbl@cs.bham.ac.uk School of Computer Science, University of Birmingham Birmingham B15 2TT, U.K. tel: +44 121-414-4792 http://www.cs.bham.ac.uk/~pbl From rrosebru@mta.ca Thu Mar 6 22:43:59 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 06 Mar 2008 22:43:59 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXSSR-0003Jw-Ml for categories-list@mta.ca; Thu, 06 Mar 2008 22:37:15 -0400 Content-Disposition: inline Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset="iso-8859-1" MIME-Version: 1.0 From: To: "Categories list" Subject: categories: Re: How to motivate a student of functional analysis Date: Thu, 06 Mar 2008 15:30:07 -0500 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 41 Ronnie points out the very excellent 1997 book on smooth analysis by Kriegl & Michor. In fact, not the reviewer but the authors themselves=20 originally stated the principle of=20 functional analysis "rather than" category theory. It is rather strange since much of the=20 material in the book was arrived at by very categorical means. For example, results published in Kriegl's joint work with Alfred Frolicher are basic. My dismay is reflected in my RCMP paper on Volterra, where I praise the book for its powerful=20 combination of functional analysis "and" category theory. In a related expositional choice the book claims to be about topological vector spaces, but the definition of morphism used betrays the fact that the weaker structures of bounded=20 sequences and of C-infinity paths are the actual underpinning. It would be instructive to know whether this strategy actually widened the audience in the=20 past 10 years. Bill On Thu Mar 6 10:15 , "Ronnie" sent: >You could also look at >MR1471480 (98i:58015) >Kriegl, Andreas; Michor, Peter W. >The convenient setting of global analysis. (English summary) >Mathematical Surveys and Monographs, 53. American Mathematical Society, >Providence, RI, 1997. x+618 pp. ISBN: 0-8218-0780-3 > >for which an e-version has been downloadable. However as the review says: >"the exposition is based on functional analysis rather than on category >theory; this fact will, undoubtedly, allow the subject to reach a wider >audience. " > > > > >Ronnie > > >----- Original Message ----- >From: "Robert L Knighten" RLK@knighten.org> >To: "Categories list" categories@mta.ca> >Sent: Thursday, March 06, 2008 2:37 AM >Subject: categories: How to motivate a student of functional analysis > > >> Most of the material connecting analysis and category theory seems to be >> written by specialists in category theory who have observed some of the >> ways >> that insights from category theory can be brought to bear (for example >> look on >> Math Sci Net at the many papers (co)authored by Joan Wick Pelletier.) B= ut >> here's an example which is a book on functional analysis that has a stro= ng >> use >> of categories: >> >> >> @book {MR0296671, >> AUTHOR =3D {Semadeni, Zbigniew}, >> TITLE =3D {Banach spaces of continuous functions. {V}ol. {I}}, >> NOTE =3D {Monografie Matematyczne, Tom 55}, >> PUBLISHER =3D {PWN---Polish Scientific Publishers}, >> ADDRESS =3D {Warsaw}, >> YEAR =3D {1971}, >> PAGES =3D {584 pp. (errata insert)}, >> MRCLASS =3D {46E15 (46M99)}, >> MRNUMBER =3D {MR0296671 (45 \#5730)}, >> MRREVIEWER =3D {H. E. Lacey}, >> } >> >> -- Bob >> >> -- >> Robert L. Knighten >> RLK@knighten.org >> >> >> >> >> -- >> No virus found in this incoming message. >> Checked by AVG Free Edition. >> Version: 7.5.516 / Virus Database: 269.21.4/1312 - Release Date: >> 04/03/2008 21:46 >> > > > > From rrosebru@mta.ca Fri Mar 7 15:28:07 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 07 Mar 2008 15:28:07 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXi7Q-00013C-NR for categories-list@mta.ca; Fri, 07 Mar 2008 15:20:36 -0400 Date: Thu, 06 Mar 2008 23:59:05 -0500 From: "Fred E.J. Linton" To: Categories list Subject: categories: Re: Heyting algebras and Wikipedia Mime-Version: 1.0 Message-ID: <851mcgeYM1534S09.1204865945@cmsweb18.cms.usa.net> Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 42 Vaughan has written, in part, that he would = > happen to find Excluded Middle more convenient than its negation, but > that's just me and perhaps others have had the opposite experience. Me too, for the most part(*), but as regards = > Then there are those who accept neither Excluded Middle nor its > negation, which takes us into the Hall of Mirrors that I always find > myself in when I go down this particular rabbit-hole. I find that it's NOT the case that I "accept neither" -- rather, it's that I sometimes prefer neither to accept it, nor to reject it, but to remain uncommitted. Noncommittally yours, -- Fred (*) I'm reminded of the legendary airline passenger who, faced with the stewardess's classic offer of "coffee, tea, or me," countered with: = "Any chance of some tonic water instead, please?" -- F. From rrosebru@mta.ca Fri Mar 7 15:28:07 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 07 Mar 2008 15:28:07 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXi50-0000k7-Iy for categories-list@mta.ca; Fri, 07 Mar 2008 15:18:06 -0400 From: Bas Spitters Subject: categories: Re: How to motivate a student of functional analysis Date: Fri, 7 Mar 2008 09:57:09 +0100 To: Categories list MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-2" Content-Transfer-Encoding: 7bit Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 43 Dear Andrej, Two examples that have not been mentioned before: * The use of the Giry monad in stochastic processes. This should motivate CS students as well as (functional) analysists. You can also use co-algebras here. * Gelfand's theorem: commutative C*-algebras are precisely the complex numbers in the topos of sheaves over its spectrum. This will also teach them that the axiom of choice is almost never needed in functional analysis and that there are good reasons to avoid it: E.g. continuous fields of C*-algebras. This is the fundamental work by Banaschewski and Mulvey. Bas From rrosebru@mta.ca Fri Mar 7 15:28:08 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 07 Mar 2008 15:28:08 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXi96-0001HS-W8 for categories-list@mta.ca; Fri, 07 Mar 2008 15:22:21 -0400 From: "Ronnie" To: "Categories list" Subject: categories: Re: How to motivate a student of functional analysis Date: Fri, 7 Mar 2008 10:31:18 -0000 MIME-Version: 1.0 Content-Type: text/plain;format=flowed; charset="iso-8859-1";reply-type=original Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 44 Bill is quite right on what the author's say. I'd also be glad of any of Bill's comments on the `Historical remarks on the development of smooth calculus', pp, 79-83, which seem very carefully put. It might interest people to give what seems the origin of the word `convenient category'. In my 1963 paper `Ten topologies for X x Y' (a title frivolously influenced by `Seven brides for seven brothers') I wrote in the Introduction: `It may be that the category of Hausdorff k-spaces is adequate and convenient for all purposes of topology'. `Convenient' here meant cartesian closed. One of the above ten topologies gives monoidal closed on all Hausdorff spaces. Some later writers removed the Hausdorff restrictions (I tried, but I was at that time not too good on final topologies). The current acount in `Topology and Groupoids' was influenced by Eldon Dyer. But for analysis the Kriegl-Michor comments show how the emphasis moved from k-spaces to ideas from Frohlicher, Bill and others. Ronnie ----- Original Message ----- From: To: "Categories list" Sent: Thursday, March 06, 2008 8:30 PM Subject: categories: Re: How to motivate a student of functional analysis Ronnie points out the very excellent 1997 book on smooth analysis by Kriegl & Michor. In fact, not the reviewer but the authors themselves originally stated the principle of functional analysis "rather than" category theory. It is rather strange since much of the material in the book was arrived at by very categorical means. For example, results published in Kriegl's joint work with Alfred Frolicher are basic. My dismay is reflected in my RCMP paper on Volterra, where I praise the book for its powerful combination of functional analysis "and" category theory. In a related expositional choice the book claims to be about topological vector spaces, but the definition of morphism used betrays the fact that the weaker structures of bounded sequences and of C-infinity paths are the actual underpinning. It would be instructive to know whether this strategy actually widened the audience in the past 10 years. Bill On Thu Mar 6 10:15 , "Ronnie" sent: >You could also look at >MR1471480 (98i:58015) >Kriegl, Andreas; Michor, Peter W. >The convenient setting of global analysis. (English summary) >Mathematical Surveys and Monographs, 53. American Mathematical Society, >Providence, RI, 1997. x+618 pp. ISBN: 0-8218-0780-3 > >for which an e-version has been downloadable. However as the review says: >"the exposition is based on functional analysis rather than on category >theory; this fact will, undoubtedly, allow the subject to reach a wider >audience. " > > > > >Ronnie > > >----- Original Message ----- >From: "Robert L Knighten" RLK@knighten.org> >To: "Categories list" categories@mta.ca> >Sent: Thursday, March 06, 2008 2:37 AM >Subject: categories: How to motivate a student of functional analysis > > >> Most of the material connecting analysis and category theory seems to be >> written by specialists in category theory who have observed some of the >> ways >> that insights from category theory can be brought to bear (for example >> look on >> Math Sci Net at the many papers (co)authored by Joan Wick Pelletier.) >> But >> here's an example which is a book on functional analysis that has a >> strong >> use >> of categories: >> >> >> @book {MR0296671, >> AUTHOR = {Semadeni, Zbigniew}, >> TITLE = {Banach spaces of continuous functions. {V}ol. {I}}, >> NOTE = {Monografie Matematyczne, Tom 55}, >> PUBLISHER = {PWN---Polish Scientific Publishers}, >> ADDRESS = {Warsaw}, >> YEAR = {1971}, >> PAGES = {584 pp. (errata insert)}, >> MRCLASS = {46E15 (46M99)}, >> MRNUMBER = {MR0296671 (45 \#5730)}, >> MRREVIEWER = {H. E. Lacey}, >> } >> >> -- Bob >> >> -- >> Robert L. Knighten >> RLK@knighten.org >> >> >> >> >> -- >> No virus found in this incoming message. >> Checked by AVG Free Edition. >> Version: 7.5.516 / Virus Database: 269.21.4/1312 - Release Date: >> 04/03/2008 21:46 >> > > > > -- No virus found in this incoming message. Checked by AVG Free Edition. Version: 7.5.516 / Virus Database: 269.21.4/1312 - Release Date: 04/03/2008 21:46 From rrosebru@mta.ca Fri Mar 7 15:28:08 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 07 Mar 2008 15:28:08 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXi3l-0000aT-BU for categories-list@mta.ca; Fri, 07 Mar 2008 15:16:49 -0400 Mime-Version: 1.0 (Apple Message framework v753) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: categories@mta.ca Content-Transfer-Encoding: 7bit From: Thorsten Altenkirch Subject: categories: Re: Heyting algebras and Wikipedia Date: Fri, 7 Mar 2008 09:25:55 +0000 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 45 Hi Paul, >> While I can't speak for Steve today, this remains a stumbling >> block for >> those raised to believe that rigorous mathematics would not be >> possible >> in a world where propositions such as "for all x and y there exists z >> such that x is a subset of z and y is a member of z" did not >> hold. How >> could x U {y} fail to exist and the walls of mathematics not come >> tumbling down? > > Maybe not the walls of mathematics, but what about theorems like > "every > polynomial functor on Set has a unique initial algebra whose > structure map > is an identity"? I think theorems like this are worth retaining (and > antifoundation makes even more of them). If we leave out "the structure map is the identity", I have no problem. The 2nd part seems to be rather cosmetical anyway, but in a bad sense of hiding the structure. Yes, I know you save some ink... > > I'd also like to suggest that "foundations" is being used in two very > different senses. In FoM, it's about quantifying the philosophical > risks > involved in particular formal systems and proofs, i.e. issues such as > relative consistency, omega-consistency, etc. For this purpose the > primacy of membership vs composition is quite immaterial. One could, I > suppose, make a formal theory based on composition equal in > strength (in > whatever sense) to ZF. > Exactly, the discussion is similar to the question whether the carta of human rights should be written in English or French. I don't care whether foundations are expressed in the language of predicate logic, category theory or type theory as long as they make sense (to me). Having said this I prefer the latter two, which work very well together, but this again has to do with beauty as opposed to cosmetics. > Category theory on the other hand is about fundamental algebraic > structures. I don't think it makes sense to ask "is category theory > omega-consistent?" as one can for ZF (not that anyone knows the > answer). Precisely! Cheers, Thorsten > > Paul > > > >> >> Vaughan >> >> Colin McLarty wrote: >>> Vaughan Pratt >>> Wednesday, March 5, 2008 8:32 am >>> >>> wrote, with much else: >>> >>>> On a related note, a careful reading of Max Kelly's "Basic >>>> Concepts of >>>> Enriched Category Theory" reveals that it is thoroughly grounded in >>>> Set,as I pointed out in August 2006 in my initial Wikipedia article >>>> on Max. >>>> I gave some thought to how one might eliminate Set from the >>>> treatment,without much success, and concluded that Max's judgment >>>> there was spot on. >>> >>> Without addressing this particular issue I want to say I >>> appreciate the >>> phrase in the article: "the explicitly foundational role of the >>> category >>> Set." I take it this is Vaughan's? >>> >>> Various people including Sol Feferman promote the view that if >>> you use >>> "sets" then you are admitting that you use ZF and not some >>> categorical >>> foundations. Vaughan's phrase goes aptly against that: If you use >>> sets, then you use sets, but there is no reason it cannot be on >>> categorical foundations. He does not say it *is* on categorical >>> foundations, and that is fine in the context. He reminds people >>> that it >>> *could* be. >>> >>> best, Colin >>> >>> >> >> >> > > -- > Paul Blain Levy email: pbl@cs.bham.ac.uk > School of Computer Science, University of Birmingham > Birmingham B15 2TT, U.K. tel: +44 121-414-4792 > http://www.cs.bham.ac.uk/~pbl > > This message has been checked for viruses but the contents of an attachment may still contain software viruses, which could damage your computer system: you are advised to perform your own checks. Email communications with the University of Nottingham may be monitored as permitted by UK legislation. From rrosebru@mta.ca Fri Mar 7 15:28:08 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 07 Mar 2008 15:28:08 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXi6H-0000uW-5o for categories-list@mta.ca; Fri, 07 Mar 2008 15:19:25 -0400 Date: Thu, 6 Mar 2008 22:18:05 -0600 From: "Michael Shulman" Subject: categories: Re: replacing set theory To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 46 On Thu, Mar 6, 2008 at 10:01 AM, Paul B Levy wrote: > Maybe not the walls of mathematics, but what about theorems like "every > polynomial functor on Set has a unique initial algebra whose structure map > is an identity"? I think theorems like this are worth retaining (and > antifoundation makes even more of them). I'm not familiar with that particular result, but I know other categorical proofs which use set-theoretic ideas like transfinite induction, and so cannot be detached from ZF in an obvious way. On the other hand, there is nothing intrinsically "membership-based" in transfinite induction. The problem seems to be the lack of a categorical analogue of ZF's axiom of replacement, since the sets in V_{\omega+\omega} already form a well-pointed elementary topos with a NNO. I find this especially mysterious because on the surface, replacement merely replaces a set by an isomorphic one (or at most a quotient)! One categorical analogue of replacement comes from categories of classes in algebraic set theory. That is, we move from a categorical analogue of ZF to an analogue of Godel-Bernays set theory. But it seems natural to wonder whether there could be a categorical analogue of replacement expressible solely as a property of the category Set, without reference to how it sits in a category of classes. Has anyone studied this question? Mike From rrosebru@mta.ca Fri Mar 7 15:28:08 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 07 Mar 2008 15:28:08 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXi2E-0000KS-4L for categories-list@mta.ca; Fri, 07 Mar 2008 15:15:14 -0400 Date: Thu, 6 Mar 2008 23:04:19 -0800 From: Toby Bartels To: categories@mta.ca Subject: categories: Categorial foundations (Was: Heyting algebras and Wikipedia) MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 47 Paul B Levy wrote in part: >what about theorems like "every >polynomial functor on Set has a unique initial algebra whose structure map >is an identity"? I think theorems like this are worth retaining (and >antifoundation makes even more of them). This is so fundamental that I'm inclined to make it an axiom. (Well, we can leave uniquiness --up to isomorphism, you mean-- and the invertibility of the structure map for theorems.) This is essentially an axiom of the Calculus of Inductive Constructions, which (like most modern type theory) is easily put in categorial language. >I'd also like to suggest that "foundations" is being used in two very >different senses. In FoM, it's about quantifying the philosophical risks >involved in particular formal systems and proofs, i.e. issues such as >relative consistency, omega-consistency, etc. For this purpose the >primacy of membership vs composition is quite immaterial. All the same, I find these matters much easier to understand when I think about them in terms of categories of sets, rather than in terms of (models of a) membership-based set theory. I would be able to read FoM if it weren't so hostile to this (although I'll follow Vaughn in noting that I haven't looked lately, so I can't speak for what it's like now). >Category theory on the other hand is about fundamental algebraic >structures. I don't think it makes sense to ask "is category theory >omega-consistent?" as one can for ZF (not that anyone knows the answer). No, but one can ask of a topos with a natural-numbers object N (and satisfying other properties that match various axioms of ZF), given a morphism X -> N whose pullbacks 0, 1, 2, ...: 1 -> N are all occupied (so each pullback has a morphism from 1), whether the negation of X over N (the internal hom [0, X] taken in the slice category over N) can also be occupied. --Toby From rrosebru@mta.ca Fri Mar 7 17:09:43 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 07 Mar 2008 17:09:43 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXjms-0006re-SI for categories-list@mta.ca; Fri, 07 Mar 2008 17:07:31 -0400 Date: Fri, 7 Mar 2008 11:07:59 -0800 From: Toby Bartels To: categories@mta.ca Subject: categories: Re: Categorial foundations MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 48 I wrote in part: >[...] ask of a topos with [...] >whether the negation of X over N (the internal hom [0, X] >taken in the slice category over N) can also be occupied. I have one and a half things backwards here. First of all, of course negation is [X, 0] rather than [0, X]. (But in exponential notation, it is 0^X; that is my excuse.) Also, my placement of "can" implies that the relevant question is whether there ~exists~ a topos E (with given properties) and there exists an object X in E (with the properties that I described); rather, the question is whether for ~every~ E (with given properties) there exists an object X in E (with the properties that I described). Iff so, then the properties required of E are omega-inconsistent. (Iff E must be a terminal category, then they are simply inconsistent. Thus omega-inconsistency is weaker than inconsistency, and omega-consistency is stronger than mere consistency.) --Toby From rrosebru@mta.ca Fri Mar 7 17:10:26 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 07 Mar 2008 17:10:26 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXjpP-0007AX-HW for categories-list@mta.ca; Fri, 07 Mar 2008 17:10:07 -0400 Date: Fri, 7 Mar 2008 11:37:15 -0800 From: Toby Bartels To: Categories list Subject: categories: Re: How to motivate a student of functional analysis MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 49 Bill Lawvere wrote in part: >Students might wonder why contiuous linear operators >are traditionally called "bounded" (when they are not even). Then Jeff Egger wrote in part: >there's certainly alot that can be said about the >category of Banach spaces and linear contractions and: >forgetful functors don't have to be "the obvious thing"; Indeed, the "obvious" forgetful functor from Ban to Set (or Top or Met) takes a Banach space to its space of all points, while the "good" one takes the space to its unit ball. Anyway, if you mix these, then a linear transformation is bounded iff it is bounded as a function from the unit ball to the space of all points. Similarly for compact linear transformations (the image is compact). That may not be the origin of these terms, but it's how I understand them. More related to category theory itself: Jeff also wrote: >in category theory, the meaning of isomorphism is fixed I'd say that the meaning of a term like "Banach space" necessarily includes the idea of what an isomorphism of such is. If different definitions define equivalent groupoids (or equivalent omega-groupoids in the most general case, as with different definitions of n-category, for example), then we can consider them equivalent defintions. So to define the essence of what Banach spaces are, one must specify (up to equivalence) the groupoid Ban_0 of Banach spaces and linear isometries between them. That said, there is some sense in the category Ban_b of Banach spaces and bounded linear transformations between them, but it is only a secondary notion compared to Ban_0. To be useful at all, it needs some extra structure, such as (at least) the dagger operator (giving duals of morphisms); then the actual isomorphisms of Banach spaces (those in Ban_0) are only the ~unitary~ (dual = inverse) isomorphisms in Ban_b. (In contrast, the category Ban as Jeff defined it needs no extra structure to be a sensible concept, since all of its isomorphisms are in Ban_0 already.) This dagger operator is used, for example, to make Hilb_b (the full subcategory of Ban_b whose objects are Hilbert spaces) into a 2-Hilbert space (from John Baez's HDA4), which is useful if you want examples of 2-Hilbert spaces; but the ~essence~ of what Hilbert spaces are is given by the groupoid Hilb_0 of linear isometries. So here is another lesson of category theory, to be taken together with Jeff's lesson last quoted above: Sometimes different notions of morphism are useful for different purposes. --Toby From rrosebru@mta.ca Fri Mar 7 17:11:43 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 07 Mar 2008 17:11:43 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JXjqc-0007NS-Pm for categories-list@mta.ca; Fri, 07 Mar 2008 17:11:23 -0400 Mime-Version: 1.0 (Apple Message framework v752.2) To: categories@mta.ca From: Steve Awodey Subject: categories: Re: replacing set theory Date: Fri, 7 Mar 2008 21:52:39 +0100 Content-Transfer-Encoding: 7bit Content-Type: text/plain;charset=US-ASCII;delsp=yes;format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 50 On Mar 7, 2008, at 5:18 AM, Michael Shulman wrote: > > > One categorical analogue of replacement comes from categories of > classes in > algebraic set theory. That is, we move from a categorical analogue > of ZF > to an analogue of Godel-Bernays set theory. But it seems natural > to wonder > whether there could be a categorical analogue of replacement > expressible > solely as a property of the category Set, without reference to how > it sits > in a category of classes. Has anyone studied this question? yes: Carsten Butz, Thomas Streicher, Alex Simpson and I did. See the first two items under 2007 on the AST site: http://www.phil.cmu.edu/projects/ast/ The short answer is, it depends on how "Sets" sits in the category of classes. In fact, *any* topos can occur as a category of "Sets" satisfying replacement in a suitable category of classes constructed from the topos. Steve Awodey From rrosebru@mta.ca Sat Mar 8 15:58:07 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 08 Mar 2008 15:58:07 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JY4yU-0000XD-1R for categories-list@mta.ca; Sat, 08 Mar 2008 15:44:54 -0400 Date: Fri, 7 Mar 2008 14:39:24 -0800 From: Toby Bartels To: categories@mta.ca Subject: categories: Re: Categorial foundations MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 51 I wrote in part: >given a morphism X -> N whose pullbacks 0, 1, 2, ...: 1 -> N >are all occupied [...] Another typo; the word "along" is missing; it should be >given a morphism X -> N whose pullbacks along 0, 1, 2, ...: 1 -> N >are all occupied [...] So: Given a collection of conditions on a locally cartesian-closed category E with an initial object 0, a final object 1, and a natural-numbers object N (various refinements should be possible for more general theories), these conditions are _omega-inconsistent_ if in every such E there exists an object X and a morphism p: X -> N such that: * defining the numerals [i]: 1 -> N using the stucture maps of N (so [0]: 1 -> N, [1]: 1 -> N -> N, [2]: 1 -> N -> N -> N, etc) and letting X_i be the pullback of p: X -> N and [i]: 1 -> N, each X_i has a morphism a_i: 1 -> X_i; * letting [X,0]_N be the internal hom from X to 0 in the slice category E/N, there is (in E itself) a morphism b: 1 -> [X,0]_N. I've read this 5 times, in different orders, so there should be no mistakes. I apologise for any confusion from my abbreviations and corrections. --Toby From rrosebru@mta.ca Sat Mar 8 15:58:07 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 08 Mar 2008 15:58:07 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JY4zZ-0000cj-DT for categories-list@mta.ca; Sat, 08 Mar 2008 15:46:01 -0400 Date: Fri, 7 Mar 2008 17:51:50 -0600 From: "Michael Shulman" Subject: categories: Re: replacing set theory To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 52 On Fri, Mar 7, 2008 at 2:52 PM, Steve Awodey wrote: > http://www.phil.cmu.edu/projects/ast/ > > The short answer is, it depends on how "Sets" sits in the category > of classes. > In fact, *any* topos can occur as a category of "Sets" > satisfying replacement in a suitable category of classes constructed > from the topos. Very interesting! But I don't think that is the answer to the question I intended to ask, although perhaps I phrased the question poorly. As far as I can tell, you give a way of interpreting replacement/collection in such a way that it is satisfied in all toposes, by "constructivizing" the existential quantifier. But as you say, "In consequence, the standard arguments using Replacement that take one outside of V_\lambda(A) for \lambda non-inaccessible, are not reproducible." What I would really like to know is, can one formulate an elementary property of a topos which *does* allow one to reproduce the standard arguments of Replacement? Here's another way to phrase the same (or a similar) question. Suppose I meet a mathematician who thinks categorically enough to dislike the membership-based nature of ZF(C), but doesn't want to give up any of its consequences. In particular, he wants to be able to use transfinite induction beyond \omega+\omega. For instance, he wants Borel determinacy to be true, which is provable in ZFC but not in Zermelo set theory (ZFC minus Replacement). Is there a categorical foundation I can tell him to use? That is, is there an elementary categorical theory which is as strong as ZF(C)? Mike From rrosebru@mta.ca Sat Mar 8 15:58:07 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 08 Mar 2008 15:58:07 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JY50F-0000gM-3I for categories-list@mta.ca; Sat, 08 Mar 2008 15:46:43 -0400 Date: Fri, 07 Mar 2008 16:07:42 -0800 From: Vaughan Pratt MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Heyting algebras and Wikipedia Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 53 Paul Levy wrote: > Maybe not the walls of mathematics, but what about theorems like "every > polynomial functor on Set has a unique initial algebra whose > structure map is an identity"? I think theorems like this are worth retaining (and > antifoundation makes even more of them). I'm not sure what "retaining" means here. Does the category Set, even as a cartesian category, have *any* properties that are open to debate? (Other than by intuitionists, who seem to thrive on quicksand.) For Set as a cartesian closed category I can see room for debate about the number of nonisomorphic sets that can appear along a chain of monics from X to 2^X when X is infinite (defined say as admitting endomonics that are not automorphisms), but relatively little in practical mathematics seems to hinge on the outcome. Correct me if I'm wrong, but my impression is that for any given language in which to express properties of Set, whether that of categories, cartesian categories, cartesian closed categories, or toposes, the properties of Set, understood classically and up to equivalence, are essentially fixed modulo largely irrelevant minutiae such as the above. Your example is a perfectly identifiable property of any category with polynomial functors (suitably defined) such as Set. If the polynomial functors are those generated from the identity functor by binary product and coproduct, e.g. X, X+X, X^2, X+X^2, etc. then it holds of Set because then the initial algebra is always the empty set (but you probably had the empty product 1 in mind as well). If Set didn't have that property it wouldn't be Set, just as Z wouldn't be Z if integer addition wasn't commutative. The property P = "for all objects x and y there exists a set z for which x is a subset of z and y is a member of z" holds of all models of ZF. It cannot be said to hold of the category Set however, not because we can't prove it, i.e. can't imagine how assuming it false could do any harm, but because we can't define it, i.e. can't imagine how it could be either true or false. What does it even mean when applied to Set as a category, or as a cartesian closed category, or even as a topos? More structure than that has to be added to Set to make P meaningful. The same goes for AFA. Chapters 1-6 of Aczel are developed starting within the ZF framework. Categories enter at Chapter 7, but Set is already fully encumbered at that point with all the machinery necessary to interpret all sentences of the language of ZF, where P is true. In that sense even FA (the Foundation Axiom) creates properties of Set that are not meaningful for Set as a mere topos. AFA as a weakening of FA means that generically there are fewer properties than with FA, not more. Fixing a particular model of AFA creates properties specific to the model, which may or may not contradict FA. (Every model of ZF is automatically a model of ZF-FA+AFA.) Vaughan From rrosebru@mta.ca Sat Mar 8 15:58:07 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 08 Mar 2008 15:58:07 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JY53C-0000vE-W3 for categories-list@mta.ca; Sat, 08 Mar 2008 15:49:47 -0400 From: Thomas Streicher Subject: categories: Re: replacing set theory To: categories@mta.ca Date: Sat, 8 Mar 2008 03:42:09 +0100 (CET) MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 54 Augmenting Steve Awodey's reply to M. Shulman I want to mention a further possibility which is more in the spirit of type / category theory, namely that of universes in toposes as described in my article with the same title (available under (www.mathematik.tu-darmstadt.de/~streicher/NOTES/UniTop.ps.gz). It is essentially a catgorical variant of Martin-Loef's notion of universe albeit an impredicative one. It was used a lot in categorical semantics of type theory (starting ~1985) but certainly part of the categorical folklore. The first written account I know of is Jean B'enabou's "Probl`emes dans le topos" from 1973. His main example that time was decidable K-finite objects in a topos with nno. A universe in a topos EE is a pullback stable class SS of morphism admitting a generic element in SS, i.e. a map E -> U in SS from which all other maps in SS can be obtained via pullback. Replacement is modelled by the requirement that SS be closed under composition. Of course, one usually requires more further closure properties (as in type theory). In the above mentioned paper I have shown that all Grothendieck and realizability toposes admit such universes (exploiting Grothendieck universes on the meta-level). As far as I can see universes serve well the purpose of replacement in mathematics, namely defining families of types by recursion. They achieve this goal in a more direct way than replacement does. The reason why they are presumably weaker than the setting Steve mentioned is that one needs a type-theoretic collection axiom (as in Joyal and Moerdijk's "Algebraic Set Theory") besides W-types for constructing set theoretic universes from type theoretic ones. I don't know why universes have hardly been considered in topos theory. (One notable exception being B'enabou's calibrations giving notions of size when considering locally small fibrations.) I think they are most useful and actually indispensible for doing category theory in the internal language of a topos. Algebraic Set Theory is an instance of universes, namely universes within categories modelling first order logic. Thomas From rrosebru@mta.ca Sat Mar 8 15:58:07 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 08 Mar 2008 15:58:07 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JY516-0000ke-0Q for categories-list@mta.ca; Sat, 08 Mar 2008 15:47:36 -0400 Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: categories@mta.ca Content-Transfer-Encoding: 7bit From: Steve Awodey Subject: categories: Re: replacing set theory Date: Sat, 8 Mar 2008 01:09:25 +0100 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 55 On Mar 8, 2008, at 12:51 AM, Michael Shulman wrote: > On Fri, Mar 7, 2008 at 2:52 PM, Steve Awodey wrote: >> http://www.phil.cmu.edu/projects/ast/ >> >> The short answer is, it depends on how "Sets" sits in the category >> of classes. >> In fact, *any* topos can occur as a category of "Sets" >> satisfying replacement in a suitable category of classes constructed >> from the topos. > > Very interesting! But I don't think that is the answer to the > question I intended to ask, although perhaps I phrased the question > poorly. As far as I can tell, you give a way of interpreting > replacement/collection in such a way that it is satisfied in all > toposes, by "constructivizing" the existential quantifier. no, the existential quantifier has its standard (categorical) interpretation (direct image), not the "constructive" one from type theory. We do not reinterpret replacement/collection either -- they have their usual interpretation. What is a bit delicate is the background category of classes in which the (set-theoretically) unbounded quantifiers are interpreted. > But as you > say, "In consequence, the standard arguments using Replacement that > take one outside of V_\lambda(A) for \lambda non-inaccessible, are not > reproducible." What I would really like to know is, can one formulate > an elementary property of a topos which *does* allow one to reproduce > the standard arguments of Replacement? > > Here's another way to phrase the same (or a similar) question. > Suppose I meet a mathematician who thinks categorically enough to > dislike the membership-based nature of ZF(C), but doesn't want to give > up any of its consequences. In particular, he wants to be able to use > transfinite induction beyond \omega+\omega. For instance, he wants > Borel determinacy to be true, which is provable in ZFC but not in > Zermelo set theory (ZFC minus Replacement). Is there a categorical > foundation I can tell him to use? That is, is there an elementary > categorical theory which is as strong as ZF(C)? > AST? Steve > Mike > From rrosebru@mta.ca Sat Mar 8 19:27:10 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 08 Mar 2008 19:27:10 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JY8OM-0007jT-SY for categories-list@mta.ca; Sat, 08 Mar 2008 19:23:50 -0400 From: Colin McLarty To: categories@mta.ca Date: Sat, 08 Mar 2008 15:45:17 -0500 MIME-Version: 1.0 Subject: categories: Re: replacing set theory Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 56 Michael Shulman Saturday, March 8, 2008 3:05 pm Asks > What I would really like to know is, can one formulate > an elementary property of a topos which *does* allow one to reproduce > the standard arguments of Replacement? Yes, What you do is start with ETCS, and adjoin an axiom scheme of replacement. The axiom scheme says: Suppose a formula associates to each element x of a set S a set we may call Sx (unique up to isomorphism). Then there is some function f:X-->S such that the fiber of f over each element x is isomorphic to Sx. Lawvere's ETCS plus this axiom scheme is intertanslateable in the obvious way with ZF, preserving theorems in both directions (you may include AxCh in both ETCS and ZF, or exclude it from both). This has been known from the earliest days of categorical set theory. My favorite early published proof was stated slightly differently, using reflection rather than replacement, but it trivially comes to the same thing. That is: AUTHOR = "Osius, Gerhard", TITLE = "Logical and set-theoretical tools in elementary topoi", BOOKTITLE = "Model Theory and Topoi", series = "Lecture Notes in Mathematics 445", PUBLISHER = "Springer-Verlag", YEAR = "1975", editor = "F. Lawvere, and C. Maurer, and G. Wraith", pages = "297--346", > Suppose I meet a mathematician who thinks categorically enough to > dislike the membership-based nature of ZF(C), but doesn't want to give > up any of its consequences. In particular, he wants to be able to use > transfinite induction beyond \omega+\omega. For instance, he wants > Borel determinacy to be true, which is provable in ZFC but not in > Zermelo set theory (ZFC minus Replacement). Is there a categorical > foundation I can tell him to use? That is, is there an elementary > categorical theory which is as strong as ZF(C)? The proof in Osius (and several later tests) implies that every consistent extension of a certain very weak fragment of ZF is intertranslateable (preserving theorems) with a corresponding extension of a certain slight extension of ETCS. Further, when you read the proof, you see the correspondence is entirely natural. For details on replacement (as opposed to Osius's use of reflection) and foundational discussion see AUTHOR = "McLarty, Colin", TITLE = "Exploring Categorical Structuralism", JOURNAL = "Philosophia Mathematica", YEAR = "2004", pages = "37--53", best, Colin From rrosebru@mta.ca Sun Mar 9 17:58:24 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 09 Mar 2008 17:58:24 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JYSMX-0001kj-9I for categories-list@mta.ca; Sun, 09 Mar 2008 17:43:17 -0300 Mime-Version: 1.0 (Apple Message framework v753) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: Categories list Content-Transfer-Encoding: 7bit From: Steve Vickers Subject: categories: Re: Heyting algebras and Wikipedia Date: Sun, 9 Mar 2008 13:43:44 +0000 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 57 On 5 Mar 2008, at 09:11, Vaughan Pratt wrote: > ... > > I have had little luck absorbing the logic of Heyting algebras into my > own mathematical thinking. I furthermore worry that if ever I were to > succeed my insights might become even less penetrating than they > already > are. > > ... > My feeling about these recommended Brouwerian modes of thoughts is > that > they are something like locker room accounts of social and other > conquests: great stories about things that never actually happened, > but > which with sufficient repetition convince one that they must surely > have > occurred. ... > > The self-evident is merely an hypothesis that is so convenient, and > that > has been assumed for so long, that we can no longer imagine it false. > This is just as true for Excluded Middle itself as for its > negation. I > happen to find Excluded Middle more convenient than its negation, but > that's just me and perhaps others have had the opposite experience. Dear Vaughan, Let me tell you how it really did occur for me - and I am happy to proclaim this as true love, not locker-room boasting. As you know from my book, at the end of the 80s I had learned from Abramsky and Smyth, not to mention the topos-theorists, that frames and topological spaces could be used to represent observational theories (technically, propositional geometric theories). But all my thinking was classical, even though I knew that frames could embody non-classical logic. I assumed that one manipulated the frames within a classical world. I was investigating how one might understand predicate geometric logic in a similar observational way, and this led me to my bagtopos construction in "Geometric Theories and Databases". But even there, I was thinking of formal logical manipulations in a classical world. In particular, I was thinking of geometric morphisms as being defined by formal translations of symbols to geometric formulae, similar to the way I treated locale maps in my book. It was Peter Johnstone who showed me a different way, with his paper "Partial products, bagdomains and hyperlocal toposes". He generalized my bagtoposes and described a universal property of them. He also showed that his more general construction was, in the contexts where mine was defined, equivalent to it. This equivalence involved describing geometric morphisms between different classifying toposes, and at that point I was expecting to see formal logical transformations. But when I eventually understood his proof I saw that he was doing something different and much more natural: he used the internal mathematics of the toposes to show how models of the geometric theories transform. This only works if the reasoning is geometric, and from then on I have grown to love the geometric reasoning better as a route to better understanding of toposes (and locales too, for that matter). This is what I tried to explain in "Locales and Toposes as Spaces", my chapter in the Handbook of Spatial Logics. As a parable, I think of toposes as gorillas (rather that elephants). At first they look very fierce and hostile, and the locker-room boasting is all tales of how you overpower the creature and take it back to a zoo to live in a cage - if it's lucky enough not to have been shot first. When it dies you stuff it, mount it in a threatening pose with its teeth bared and display it in a museum to frighten the children. But get to know them in the wild, and gain their trust, then you begin to appreciate their gentleness and can play with them. The gorilla in the cage is the topos in the classical world. Best regards, Steve. Here's something that tickled me. A sign outside a monastery in Meteora, Greece, says - "O topos einai ieros" ("The topos is holy") From rrosebru@mta.ca Mon Mar 10 12:57:10 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 10 Mar 2008 12:57:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JYkHL-0000fL-M8 for categories-list@mta.ca; Mon, 10 Mar 2008 12:51:07 -0300 Date: Mon, 10 Mar 2008 10:14:35 +0100 (CET) From: Elvira Albert To: ppdp08-cfp@clip.dia.fi.upm.es Subject: categories: PPDP 2008 - 2nd Call for Papers MIME-Version: 1.0 Content-Type: TEXT/PLAIN; CHARSET=ISO-8859-1 Content-Transfer-Encoding: QUOTED-PRINTABLE Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 58 =2E............................................................... ACM PPDP 2008 - Call For Papers 10th ACM-SIGPLAN International Symposium on Principles and Practice of Declarative Programming Valencia, Spain, July 15-17, 2008 http://www.clip.dia.fi.upm.es/Conferences/PPDP08 =2E............................................................... IMPORTANT DATES Submission: April 10, 2008 Notification: May 15, 2008 Conference: July 15-17, 2008 SCOPE: PPDP 2008 is a forum for researchers and practitioners in the declarative programming communities. It solicits papers on all aspects of logic, constraint and functional programming, as well as on related paradigms such as visual programming, executable specification languages, database languages, AI and knowledge representation languages for the "semantic web". MAIN TOPICS: Logic, Constraint, and Functional Programming; Database, AI and Knowledge Representation Languages; Visual Programming; Executable Specification for Languages; Applications of Declarative Programming; Methodologies for Program Design and Development; Declarative Aspects of Object-Oriented Programming; Concurrent Extensions to Declarative Languages; Declarative Mobile Computing; Paradigm Integration; Proof Theoretic and Semantic Foundations; Type and Module Systems; Program Analysis and Verification; Program Transformation; Abstract Machines and Compilation; Programming Environments. PROCEEDINGS: The proceedings will be published by ACM Press RELATED EVENTS: PPDP 2008 will be co-located with the 15th International Static Analysis Symposium (SAS 2008) and the 18th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2008). SYMPOSIUM CHAIR: Elvira Albert, Complutense University of Madrid PROGRAM CHAIR: Sergio Antoy, Portland State University INVITED SPEAKER: Michael Leuschel, University of D=C3=BCsseldorf, Germany PROGRAM COMMITTEE: Elvira Albert Complutense University of Madrid, Spain Sergio Antoy Portland State University, USA Maribel Fernandez King's College London, UK Maurizio Gabbrielli University of Bologna, Italy Neil Ghani University of Nottingham, UK Masami Hagiya University of Tokyo, Japan Joxan Jaffar National University, Singapore Claude Kirchner INRIA Bordeaux, France Herbert Kuchen University of Muenster, Germany Michael Maher NICTA and University of New South Wales, Australia Dale Miller INRIA Saclay, France Eugenio Moggi University of Genova, Italy Kostis Sagonas Uppsala University, Sweden Carsten Schurmann, IT University of Copenhagen, Denmark Peter Sestoft IT University of Copenhagen, Denmark LOCAL CHAIR: Christophe Joubert From rrosebru@mta.ca Mon Mar 10 12:57:10 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 10 Mar 2008 12:57:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JYkKB-0000zg-Oe for categories-list@mta.ca; Mon, 10 Mar 2008 12:54:03 -0300 From: "Bhupinder Singh Anand" To: "'Categories list'" Subject: categories: Re: Heyting algebras and Wikipedia Date: Mon, 10 Mar 2008 16:12:57 +0530 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 59 On 5 Mar 2008, at 09:11, Vaughan Pratt wrote: VP>> My feeling about ... Brouwerian modes of thoughts is that they are = ... great stories about things that never actually happened ... I happen to = find Excluded Middle more convenient than its negation, but that's just me = and perhaps others have had the opposite experience. < Envelope-to: categories-list@mta.ca Delivery-date: Mon, 10 Mar 2008 21:42:06 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JYsMh-0004HN-Nj for categories-list@mta.ca; Mon, 10 Mar 2008 21:29:11 -0300 From: MFPS To: categories@mta.ca Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit Mime-Version: 1.0 (Apple Message framework v919.2) Subject: categories: MFPS Deadline This Friday Date: Mon, 10 Mar 2008 17:52:11 -0500 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 60 Dear Colleagues, This is a reminder that full submissions for MFPS are due this Friday, March 14. We do NOT require titles and abstracts to have been submitted for full submissions to be considered for presentation at the meeting. Submissions are through EasyChair and can be made at http://www.easychair.org/conferences/?conf=mfps24 Full details about the conference are available at http://www.math.tulane.edu/~mfps/mfps24.htm Best regards, Mike Mislove Mathematical Foundations of Programming Semantics http://www.math.tulane.edu/~mfps From rrosebru@mta.ca Tue Mar 11 19:56:20 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 11 Mar 2008 19:56:20 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JZDGs-0003Yj-3C for categories-list@mta.ca; Tue, 11 Mar 2008 19:48:34 -0300 Date: Tue, 11 Mar 2008 00:33:15 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: Categories list Subject: categories: Re: Heyting algebras and Wikipedia Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 61 Steve Vickers wrote: > The gorilla in the cage is the topos in the classical world. According to http://en.wikipedia.org/wiki/Gorilla, gorillas in captivity tend to obesity and earlier maturation of females, and have been taught sign language. It doesn't mention any other differences, and the rest of the article is about gorillas in the wild. Does this make a "topos in the classical world" a gros topos and the wild ones petit? The analogy is rather on the colorful side for me. The first half of http://en.wikipedia.org/wiki/Topos is about Grothendieck topoi [sic], the rest about elementary toposes (was topoi but I changed it out of deference to PTJ's strong feelings in the matter). The Explanation section (my contribution, intended as a response to the "respectful awe" tone of the comments on the article's talk page reacting to the bald definition, i.e. the commenters seemed largely mystified but accepted this as par for the course for anything this far beyond rocket science) is presented from the point of view of elementary toposes as a solution to the problem of characterizing the notion of subobject in elementary terms. My question to Steve and the list as a whole would be, if you had been assigned the task of writing an explanation section following the formal definition section, where would you have put the emphasis: on how the definition facilitates a first-order characterization of the notion of "subobject", or on the geometric morphism perspective? Vaughan From rrosebru@mta.ca Wed Mar 12 08:46:57 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 12 Mar 2008 08:46:57 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JZPFD-0002em-G8 for categories-list@mta.ca; Wed, 12 Mar 2008 08:35:39 -0300 MIME-Version: 1.0 Subject: categories: Re: Heyting algebras and Wikipedia Date: Wed, 12 Mar 2008 08:58:19 -0000 From: "Townsend, Christopher" To: Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 62 Vaughen wrote: My question to Steve and the list as a whole would be, if you had been assigned the task of writing an explanation section following the formal definition section, where would you have put the emphasis: on how the definition facilitates a first-order characterization of the notion of "subobject", or on the geometric morphism perspective? My answer: Topos theory, like other theories, is about the interaction of objects (toposes) and morphisms (geometric morphisms). I see it as inherently an aspect of category theory. We use category theory to define it and I believe the most effective expositions on topos theory are those that are based on category theory. Agreed, a strength of the definition of topos is that it allows a multi-faceted approach (three blind men ...) but my personal view is that this must confuse any newcomer. A topos is a type of category which [insert definition here]. Some of the basic results of topos theory are [insert categorical lemmas here]. Once these categorical foundations are in place one is able to (a) investigate/research toposes and (b) learn more about other (say logical) aspects of toposes. =20 I don't want to detract from the importance of (b), but (a) can be carried out without (b). Christopher =20 -----Original Message----- From: cat-dist@mta.ca [mailto:cat-dist@mta.ca] On Behalf Of Vaughan Pratt Sent: 11 March 2008 07:33 To: Categories list Subject: categories: Re: Heyting algebras and Wikipedia Steve Vickers wrote: > The gorilla in the cage is the topos in the classical world. According to http://en.wikipedia.org/wiki/Gorilla, gorillas in captivity tend to obesity and earlier maturation of females, and have been taught sign language. It doesn't mention any other differences, and the rest of the article is about gorillas in the wild. Does this make a "topos in the classical world" a gros topos and the wild ones petit? The analogy is rather on the colorful side for me. The first half of http://en.wikipedia.org/wiki/Topos is about Grothendieck topoi [sic], the rest about elementary toposes (was topoi but I changed it out of deference to PTJ's strong feelings in the matter). The Explanation section (my contribution, intended as a response to the "respectful awe" tone of the comments on the article's talk page reacting to the bald definition, i.e. the commenters seemed largely mystified but accepted this as par for the course for anything this far beyond rocket science) is presented from the point of view of elementary toposes as a solution to the problem of characterizing the notion of subobject in elementary terms. My question to Steve and the list as a whole would be, if you had been assigned the task of writing an explanation section following the formal definition section, where would you have put the emphasis: on how the definition facilitates a first-order characterization of the notion of "subobject", or on the geometric morphism perspective? Vaughan From rrosebru@mta.ca Wed Mar 12 19:53:23 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 12 Mar 2008 19:53:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JZZl9-00040y-GI for categories-list@mta.ca; Wed, 12 Mar 2008 19:49:19 -0300 Mime-Version: 1.0 (Apple Message framework v752.2) Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed To: categories@mta.ca Subject: categories: Lectureship at Sussex Date: Wed, 12 Mar 2008 15:18:38 +0000 From: Bernhard Reus Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 63 Our Department (Informatics, Sussex) is looking for a Lecturer =20 (permanent, full time) in *Foundations of Computation* . For full details and how to apply see Below you find the main text of the ad. Cheers, Bernhard ----------------------------------------------------------------------- Salary range: =A327,466 to =A340,335 pa Expected start date: 1 September 2008 or soon after The Department (graded 5 in all RAEs to date) is seeking to appoint a =20= Lecturer in Foundations of Computation. The successful applicant will =20= have high quality peer reviewed publications, relevant teaching =20 experience, and be prepared to contribute to the administrative tasks =20= of the department. The Foundations group has a strong portfolio of research in =20 developing semantic theories and mathematical models for languages =20 and systems. Applicants should have research interests in one or more =20= of the following areas: programming language theory, program logics, =20 theory of quantum computation, type theory, domain theory, =20 concurrency, or theory of pervasive/ubiquitous computing. Informal enquiries may be addressed to: Dr Ian Mackie, tel 01273 =20 873117, email I.Mackie@sussex.ac.uk; Prof John Carroll (Head of =20 Department), tel 01273 678029, email J.A.Carroll@sussex.ac.uk When completing the University application form please make sure you =20 include your CV and list of publications. Closing date for applications: 31 March 2008 Interview date: 2 May =20 2008= From rrosebru@mta.ca Wed Mar 12 19:53:24 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 12 Mar 2008 19:53:24 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JZZkJ-0003ul-Mb for categories-list@mta.ca; Wed, 12 Mar 2008 19:48:27 -0300 Date: Wed, 12 Mar 2008 13:33:25 +0000 From: Steve Vickers MIME-Version: 1.0 To: Categories list Subject: categories: Re: Heyting algebras and Wikipedia Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 64 Vaughan Pratt wrote: > > My question to Steve and the list as a whole would be, if you had been > assigned the task of writing an explanation section following the formal > definition section, where would you have put the emphasis: on how the > definition facilitates a first-order characterization of the notion of > "subobject", or on the geometric morphism perspective? Dear Vaughan, I would try to encapsulate what I said in "Locales and toposes as spaces". My aim there was to present ideas (of generalized space) that have been known to topos-theorists from the start, but which tend to get buried in the technical development. I wanted to show how results in Mac Lane and Moerdijk, or in the Elephant, can be read in a non-standard order to paint a particular picture. (To indulge myself in metaphors again, the blind toddler wanders right underneath, between the legs, feels nothing, and concludes the elephant is very like a generalized space. In its innocence, it never developed the respectful awe of the scary bits.) Some of the encapsulation is in my Linz lecture slides http://www.cs.bham.ac.uk/~sjv/LinzTalk.pdf My aim there was to explain how sheaves can be viewed as continuous set-valued maps, and what that tells us about continuity. In particular, I wanted to support the idea that "continuous" means "geometrically definable". This leads to the idea of geometric morphism between toposes as continuous map between generalized spaces. But then non-classical reasoning (no excluded middle, no axiom of choice, no impredicative constructions) enters in immediately as part of the geometricity. Best regards, Steve. From rrosebru@mta.ca Thu Mar 13 10:41:35 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 13 Mar 2008 10:41:35 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JZnST-0004jS-Pn for categories-list@mta.ca; Thu, 13 Mar 2008 10:26:58 -0300 Date: Wed, 12 Mar 2008 22:37:49 -0500 From: "Michael Shulman" Subject: categories: Re: replacing set theory To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 65 On Sat, Mar 8, 2008 at 2:45 PM, Colin McLarty wrote: > > What I would really like to know is, can one formulate an elementary > > property of a topos which *does* allow one to reproduce the standard > > arguments of Replacement? > > Yes, What you do is start with ETCS, and adjoin an axiom scheme of > replacement. [...] Thank you! This is exactly what I was looking for. > This has been known from the earliest days of categorical set > theory. But it doesn't seem to be *well* known any more, or at least well-disseminated and exposited. Several people have told me that they didn't think it was possible to express replacement category-theoretically without using a category of classes. And even now knowing what I'm looking for, I am unable to find more than a sentence or two about it in any book on topos theory, none of which actually gives any version of the axiom. > AUTHOR = "McLarty, Colin", > TITLE = "Exploring Categorical Structuralism", This raises another question. You mention at the end of this paper that large-cardinal axioms are "routinely pursued in isomorphism-invariant terms". This is clear to me for many types of large cardinals, but not for the stronger ones that involve elementary embeddings of the universe of sets. Ultrapowers have a categorical analogue, of course (filterquotient) but then there is a transitive collapse of the entire universe, from which I don't see immediately how to eliminate the global membership predicate. Can you give a clue or a reference? Thanks again, Mike From rrosebru@mta.ca Thu Mar 13 22:13:29 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 13 Mar 2008 22:13:29 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JZyNi-00030u-6N for categories-list@mta.ca; Thu, 13 Mar 2008 22:06:46 -0300 Content-Type: text/plain; charset=US-ASCII; format=flowed Content-Transfer-Encoding: 7bit From: Paul Taylor Subject: categories: mathematical articles in online encyclopedias Date: Thu, 13 Mar 2008 16:31:36 +0000 To: Categories list Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 66 It was Vaughan Pratt who first introduced the Wikipedia thread, in response to someone who said that he hadn't heard of Heyting algebras. I added my penniworth, having in mind its treatment of Dedekind cuts and Locally compact spaces. Recently, however, discussion has centred on the notion of Topos. I have changed the Subject line because Wikipedia is not the only site of its kind. Anyone thinking of writing for it should perhaps also consider: - citizendium.org - which looks like Wikpedia because it is run by the latter's co-founder and now unperson; citizendium has a strict policy of using real names and qualifications; - planetmath.org - in which authors "own" the pages that they have written until they've demonstrably abandoned them; - mathworld.wolfram.com - beware that this is owned by Wolfram. It seems to me that toposes are not a good example on which to base this discussion, being too advanced a topic. On the other hand, if you have opinions about what Wikipedia and the other sites should say about them, then go ahead and write your article, instead of discussing it here! But before you write, please bear in mind that these are resources for the general educated user, not for specialists in the field. Have a look around for articles that you find informative about completely different subjects, for example a medical topic or a place of interest. It should begin by making sure that the reader has come to the right place, for example the word "topos" is also used about poetry. Then it should tell the lay person why anybody would spend their time thinking about this thing. As we all know, a topos is an elephant. Its trunk looks like constructive set theory, its legs look like topological spaces and its tail is a group. But I really don't think that a specialist in a particular topic in mathematics should be writing about their speciality. You need to see it from a distance. Wikipedia has policies forbidding original research. Encyclopedia articles should provide "general knowledge" about background topics. My research programme is a reformulation of general topology, so I need to talk about this against a background of general knowledge about traditional point-set topology, locale theory, continuous lattices, domain theory, constructive analysis and so on. However, since I am doing something completely new, I really don't want to have to give an account of these subjects before I say my own stuff, so I would like to be able to cite a textbook or other source that does so. And I would like that source to be accessible to student without specialist knowledge, and NOT depend on or be part of some other partisan presentation. Chances are that any account of a topic that is part of a research paper will depend on somebody else's foundational system. For example, I would like to refer to an account of locally compact spaces. Having been exposed to locale theory and continuous lattices for 25 years now, I regard it as a matter of "general knowledge" that the topologies of locally compact spaces are continuous lattices, and these in turn carry topologies, named after Jimmie Lawson and Dana Scott. However, I find NOTHING about this in the Wikipedia article. That and more or less every other article there about topological subjects adheres to the orthodox view in pure mathematics that all self- respecting topological spaces are Hausdorff. If I write a new article about locally compact spaces for Wikipedia then I will find myself in conflict both with Wikipedia's anonymous cliques and also with the mathematical establishment. If you're interested in rings and not non-Hausdorff spaces, then please substitute the commutative axiom for Hausdorffness in what I've just said. If some basic result about rings, fields or modules can be formulated without assuming commutativity or charactersistic zero, at no or a small extra cost, then surely it should be so formulated. If the more general treatment is more complicated, but throws light on the subject, the simpler one should be given first, and an overview of the more general one afterwards. Another example of this is excluded middle. Personally, I foreswore EM about 15 years ago because I was disgusted by some of the arguments that people were using in domain theory - "bit-picking", I called it - that didn't form part of any applications or philosophy. EM leads to ugly mathematical arguments, and in very many cases can simply be avoided by stating them more carefully. In others (for example intuitionistic ordinals and constructive analysis) there is a more interesting theory when you use the more delicate tools of constructive reasoning. Getting back to encyclopedias, remember that they are for teaching, not research. Tell students and the general public why the topic is interesting, and tempt them with some simple point that they may not have considered. I'm not claiming to be very good at these things myself, but there are others who regularly do so on their blogs, as well as in Wikipedia and the like. If you don't already know them, you might like to take a look at the blogs by - Andrej Bauer - math.andrej.com - John Baez et al - golem.ph.utexas.edu/category - Dan Pioni alias sigfpe - sigfpe.blogspot.com Paul Taylor From rrosebru@mta.ca Fri Mar 14 08:47:16 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 14 Mar 2008 08:47:16 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ja8JG-0001pY-Vb for categories-list@mta.ca; Fri, 14 Mar 2008 08:42:51 -0300 Mime-Version: 1.0 (Apple Message framework v752.2) Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII; format=flowed To: types-announce@lists.seas.upenn.edu, categories@mta.ca Subject: categories: Correction: Lectureship at Sussex Date: Thu, 13 Mar 2008 15:40:43 +0000 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: From: cat-dist@mta.ca Status: O X-Status: X-Keywords: X-UID: 67 Unfortunately, in my previous posting a wrong link was given. The correct link, if you're interested, is Apologies. Bernhard From rrosebru@mta.ca Fri Mar 14 08:47:16 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 14 Mar 2008 08:47:16 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ja8L3-00020K-Rk for categories-list@mta.ca; Fri, 14 Mar 2008 08:44:41 -0300 From: Thomas Streicher Subject: categories: categorical formulations of Replacement To: categories@mta.ca Date: Fri, 14 Mar 2008 03:34:12 +0100 (CET) MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 68 The Replacemnt axiom which Colin formulated in his article in Phil.Math. only works for well-pointed categories. But even in this framework it is too strong due to its requirement that every external family arises from an internal one. So it fails for example for the model of ETCS arising from a countable model of ZFC because there are only countably many internal families over N whereas there uncountable many external families indexed by (the global elements of) N. A defect of the work from the 70ies (Cole, Osius at.al.) is that it just proves equiconsistency of ETCS and bZ (bounded Zermelo set theory) and not an equivalence between models of ETCS and bZ. I think that the more interesting question is what is a model of intuitionistic set theoy which cannot be well-pointed. For this purpose it is INDISPENSIBLE to have in our category an object U of all sets. This was first recognized and formulated by Christian Maurer in his paper "Universes in Topoi" which appeared in the SLNM volume "Model theory and topoi" (ed. Lawvere, Maurer, Wraith). Maurer was working in a topos and postulated an object U (a "universe") of this topos with ext : U >-> P(U) satisfying a few axioms which ensure that U is a model for IZF (without saying so). In particular, he has a clear formulation of the axiom of replacement, namely (\forall a : U) (\forall f : U^{ext(a)}) (\exists b : U) (\forall y : U) (y \in ext(b) <-> (\exists x \in ext(a)) y = f(x)) albeit in a somewhat less readable since he avoids the internal language of the topos. This point of view was taken up later in the Algebraic Set Theory (AST) of Joyal and Moerdijk whose work concentrated on CONTRUCTING (initial) universes of this kind. A couple of years later Alex Simpson in his LiCS'99 paper took up Maurer's early insight (at least he refers to Maurer's paper) but weakened the ambient category to be a model for first order logic (as set theorists do). The main new ingredient of AST (and Alex's paper) is the assumption of a class of small maps giving (cum grano salis) a notion of "size" (like B'enabou's calibrations but satisfying much stronger axioms) together with a notion of small powerset functor P_s (depending on the class of small maps). A universe is then defined as a(n initial) fixpoint U of P_s which, of course, can't be small itself. A point which seems to have been overlooked in this latest discussion is that Replacement per set is not very strong. It gets its usual strength only in presence of unbounded separation. See the paper by Awodey, Butz, Simpson and me which appeared in last year's Bull.Symb.Logic (see also http://homepages.inf.ed.ac.uk/als/Research/Sources/set-models-announce.pdf) where we discuss this in more detail. The point is that around every topos EE one can build a category of classes whose small "set" part is equivalent to EE. The corresponding class theory BIST is thus conservative over topos logic (with nno). Later on Awodey and his students have also studied the much weaker "predicative" case where replacement still holds. See also Aczel and Rathjen's work on CZF in this context which is much older than AST dating back to articles by Aczel in the late 70ies (and based on previous work by John Myhill). Thomas Streicher From rrosebru@mta.ca Thu Mar 13 22:13:29 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 13 Mar 2008 22:13:29 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JZyNE-0002yD-RN for categories-list@mta.ca; Thu, 13 Mar 2008 22:06:16 -0300 From: Colin McLarty To: categories@mta.ca Date: Thu, 13 Mar 2008 09:52:25 -0400 MIME-Version: 1.0 Content-Language: en Subject: categories: Re: replacing set theory Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 69 Wednesday, March 12, 2008 11:37 pm Subject: Re: categories: Re: replacing set theory Writes of the replacement scheme in categorical set theory > But it doesn't seem to be *well* known any more, or at least > well-disseminated and exposited. Several people have told me that > they didn't think it was possible to express replacement > category-theoretically without using a category of classes. There are two issues here, because there are two things to mean by "replacing set theory." On one hand it can mean replacing the membership-theoretic approach of ZF by the categorical approach of ETCS. This has long been routine and full details are found in Osius "Categorical Set theory: A Characterization of the Category of Sets", (Journal of Pure and Applied Algebra, 1974, pages 79--119). People do not talk about it a lot because it was a well-focussed problem which got a perfectly good answer and at the time there more pressing and more open-ended research issues. Perhaps also because Saunders Mac~Lane tended to stress that lots of higher set theory (like lots of logic in general) is very weakly linked to most of mathematics -- which is true, but is not to say that higher set theory (or logic in general) need be abandoned. He hoped they could be brought back into better touch. This is the issue raised for example by Feferman and Rao in Giandomenico Sica ed. _What is category theory?_ {Polimetrica, 2006) when they claim it is "unclear" whether certain ZF constructions can be given at all in categorical terms. Yes, it is clear they can, proven in detail by Osius in 1976 (not the first proof but my favorite reference on it). On the other hand, for some purposes we want to replace the category Set (no matter how it is axiomatized) by something more general, often by any elementary topos, or it could be any Boolean topos, or category with a category of classes. The categorical replacement scheme I mentioned generalizes very well to any well-pointed topos, but that is little more general than Set. It makes much use of fibers over global elements. To put it in the terms I like, it does nothing to say a family of sets S-->I defined by replacement should be "smooth" or "continuous over the index set I" in anyway. In Set the index *set* is not smooth or continuous to begin with, it is a discrete set. The idea of categories of classes is to get some sense of large collections that *do* vary "smoothly" or "continuously" when the base has some smooth or continuous character (in a very general sense, so for example effective computability is the relevant "smoothness" for those types in the effective topos close to the natural numbers). The next question brings us back to the first perspective. It is about the category Set, but wants to approach that category by categorical tools. > This raises another question. You mention at the end of this paper > that large-cardinal axioms are "routinely pursued in > isomorphism-invariant terms". This is clear to me for many types of > large cardinals, but not for the stronger ones that involve elementary > embeddings of the universe of sets. Ultrapowers have a categorical > analogue, of course (filterquotient) but then there is a transitive > collapse of the entire universe, from which I don't see immediately > how to eliminate the global membership predicate. Can you give a clue > or a reference? Within ZF itself the global membership predicate X \in S is just one guise of the relation "the membership-tree of X is (isomorphic to) the restriction of the membership-tree of S to some node directly below the top." Transitive collapse can be re-cast in these terms as dealing with all well-founded extensional relations and not only dealing with restricted membership (i.e. restricted to some inner model of ZF) and actual membership of sets. In isomorphism invariant terms, a transitive collapse of the universe means a uniform method of taking any well-founded extensional relation (not just ZF membership on each transitive closure) and restricting it to a sub-relation which is still well-founded and extensional (which we do not bother collapsing to membership on some transitive closure) while preserving some properties of the first relation. And then we ask if a given collapse has any fixed points: are there any well-founded extensional relation so big that this collapse leaves an isomorphic relation? I have no idea what the practical effect would be of recasting collapse in these terms. I have never worked with large cardinals and transitive collapse. But it certainly *can* be recast this way. Some one should try it. best, Colin From rrosebru@mta.ca Thu Mar 13 22:13:29 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 13 Mar 2008 22:13:29 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JZyPr-0003D5-Vp for categories-list@mta.ca; Thu, 13 Mar 2008 22:09:00 -0300 Content-Disposition: inline Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset="iso-8859-1" MIME-Version: 1.0 From: To: categories@mta.ca Subject: categories: Re: replacing set theory Date: Thu, 13 Mar 2008 17:38:11 -0400 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 70 If one assumes that epics split and that the Boolean algebra of subsets of 1 reduces to two elements, then (to a first approximation) the main effect of axiom schemes is to provide=20 larger cardinals.=20 That is explicitly exemplified in the=20 document that has been available in the University of Chicago Math Library since 1965, and that has=20 been available in recent years as a TAC Reprint. Using the definable classes of sets smaller than any=20 given set, the postulate that there are arbitrarily large such classes closed under arbitrary definable operations phi is proposed. I am=20 not aware of any further studies of that postulate. (This may be the=20 first time that a geometer has=20 shown persistent interest in=20 replacement) Of course above I use the term "class" in the intuitive sense of a natural subset of every model of the theory ("meta-" in Mac lane's terminology),=20 that objective meaning corresponding subjectively to a formula of the theory. That is, not in the sense of a half- hearted attempt to represent classes by elements V of the model. (Less half-hearted is the proposal that seems implicit in the 1963 reaction of Goedel & Bernays=20 themselves when they heard, presumably from Kreisel, that work was underway on a=20 categorical set theory. Namely try a category of classes of classes etc satisfying at least the intuitive=20 property of cartesian closure.) The usual replacement scheme=20 does seem at first to yield only cardinals smaller than given ones.=20 But in the hierarchical view, that=20 quotient set consists of elements=20 which themselves have elements, thus the actual mathematical=20 content is that of a family of=20=20 sets, a concept whose=20 geometrical expression is that=20 of a fibration, hence Colin's=20 formulation of the axiom. Indeed the formulation goes back many years, but I don't have a reference. Concerning an elementary self-embeddings of the universe, it is in any case an additional functor added to the=20 basic structure, and since the basic structure of category is first-order, a scheme could be considered to the effect that such a functor commute with quantifiers. Bill On Wed Mar 12 23:37 , "Michael Shulman" sent: >On Sat, Mar 8, 2008 at 2:45 PM, Colin McLarty colin.mclarty@case.edu> wrot= e: >> > What I would really like to know is, can one formulate an elementary >> > property of a topos which *does* allow one to reproduce the standard >> > arguments of Replacement? >> >> Yes, What you do is start with ETCS, and adjoin an axiom scheme of >> replacement. [...] > >Thank you! This is exactly what I was looking for. > >> This has been known from the earliest days of categorical set >> theory. > >But it doesn't seem to be *well* known any more, or at least >well-disseminated and exposited. Several people have told me that >they didn't think it was possible to express replacement >category-theoretically without using a category of classes. And even >now knowing what I'm looking for, I am unable to find more than a >sentence or two about it in any book on topos theory, none of which >actually gives any version of the axiom. > >> AUTHOR =3D "McLarty, Colin", >> TITLE =3D "Exploring Categorical Structuralism", > >This raises another question. You mention at the end of this paper >that large-cardinal axioms are "routinely pursued in >isomorphism-invariant terms". This is clear to me for many types of >large cardinals, but not for the stronger ones that involve elementary >embeddings of the universe of sets. Ultrapowers have a categorical >analogue, of course (filterquotient) but then there is a transitive >collapse of the entire universe, from which I don't see immediately >how to eliminate the global membership predicate. Can you give a clue >or a reference? > >Thanks again, >Mike > > > From rrosebru@mta.ca Fri Mar 14 20:25:54 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 14 Mar 2008 20:25:54 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JaJ7P-0007C0-1e for categories-list@mta.ca; Fri, 14 Mar 2008 20:15:19 -0300 Mime-Version: 1.0 (Apple Message framework v624) Content-Type: text/plain; charset=US-ASCII; format=flowed Content-Transfer-Encoding: 7bit From: Paul Taylor Subject: categories: the axiom scheme of replacement in category theory Date: Fri, 14 Mar 2008 15:13:45 +0000 To: Categories list Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 71 Michael Shulman is quite right to press the question of how to express the axiom scheme of replacement WITHOUT using classes or universes. There are two traditions on this topic in set theory. One of them states replacement and collection "from the outside" by using classes or universes. The other, found in ZF itself, somehow builds it up "from the inside", without pre-supposing classes, although the large cardinals that result from this hypothesis can themselves be used as universes for weaker set theories. Steve Awodey and Thomas Streicher have referred to the excellent work in Algebraic Set Theory that is collected on Steve's web page at www.phil.cmu.edu/projects/ast/ ASD is a categorical treatment that follows the "sets and classes" tradition. However, Mike acknowledged that in his original posting, and, with all due respect, Steve and Thomas have NOT answered his question there: > I'm not familiar with that particular result, ie "every polynomial functor on Set has a unique initial algebra whose structure map is an identity", mentioned by Paul Levy, > but I know other categorical proofs which use set-theoretic ideas > like transfinite induction, and so cannot be detached from ZF in an > obvious way. On the other hand, there is nothing intrinsically > "membership-based" in transfinite induction. The problem seems to > be the lack of a categorical analogue of ZF's axiom of replacement, > since the sets in V_{\omega+\omega} already form a well-pointed > elementary topos with a NNO. I find this especially mysterious > because on the surface, replacement merely replaces a set by an > isomorphic one (or at most a quotient)! > One categorical analogue of replacement comes from categories of > classes in algebraic set theory. That is, we move from a categorical > analogue of ZF to an analogue of Godel--Bernays set theory. But it > seems natural to wonder whether there could be a categorical analogue > of replacement expressible solely as a property of the category Set, > without reference to how it sits in a category of classes. Has anyone > studied this question? Although he had expressed himself quite clearly the first time, it seems that he needed to put his question again: > "In consequence, the standard arguments using > Replacement that take one outside of V_\lambda(A) for \lambda > non-inaccessible, are not reproducible." What I would really like to > know is, can one formulate an elementary property of a topos which > *does* allow one to reproduce the standard arguments of Replacement? I agree that it would be very nice to have a formulation of Replacement in category theory that was as simple as, say, the notion of subobject classifier. I proposed such a thing in the final pages of my book, "Practical Foundations of Mathematics" (CUP, 1999), and this posting leads up to my formulation. Colin McLarty gave a non-AST answer to Mike's question, citing a paper by Gerhard Osius and one of his own. I confess that I have not seen either of these, not currently having access to a university library, so please excuse me if they already say what I am about to say. I would, however, ask the algebraic set theorists to acknowledge that there is another point of view on the question, albeit one that has not been worked out so extensively as theirs. Anybody who has attempted to discuss foundations with either set theorists or mainstream mathematicians knows that it is impossible to get them to engage. Colin McLarty has already pointed this out in this thread, saying that > various people including Sol Feferman promote the view that > if you use "sets" then you are admitting that you use ZF and > not some categorical foundations. Of course, set theorists who claim this are simply hijacking the English language, and we have to resist them in so doing. One problem on the other side is that most mathematicians couldn't actually quote the ZF axioms that they claim to use, without looking them up. They could, on the other hand, readily give a practical account of - cartesian products, subsets, disjoint sums and quotients of equivalence relations, which are axiomatised in category theory as a "pretopos"; - primitive recursion over N, lists and finite subsets, which, together with the above, make an "arithmetic universe"; and - full powersets, which make an "elementary topos"; because they teach this to first year mathematics and computer science undergraduates. In Section 2.2 of my book I set out what I call "Zermelo type theory", because it differs very little from his set theory. This is what ordinary mathematicians use, and it is also the ("Mitchell--Benabou internal") language of a topos. I would be quite willing to explain to any set theorist how any statement in my book (apart from the last section) can be understood in categorical terms within an elementary topos. (One of the many uses that I would have for a time machine would be to go back 100 years, tap Ernst Zermelo on the shoulder while he was writing "Untersuchungen uber die Grundlagen der Mengenlehre I", and ask him to make those pairs ORDERED, as it would have avoided so much trouble since then! Note that Bourbaki's "Theorie des Ensembles" DOES have ordered pairs.) Some time ago, Saunders Mac Lane found himself in the middle of a dispute with the set theorists because he had claimed that most of mathematics could be expressed in an elementary topos with a natural numbers object. Unfortunately, I have never managed to work out exactly which of his publications was the centre of the controversy - maybe Colin can fill us in here on the bibliographical details. The bit that's missing, as Mike said, is the axiom scheme of replacement. But we are back to the original problem of communication with set theorists in trying to identify what this actually means. They are rather like Proteus in Greek mythology, who could foretell the future, but only to someone who could capture him, which he would avoid by repeatedly changing form - into a lion, serpent, leopard, pig, water or tree (according to Google and Wikipedia). At first, Replacement appears just to provide the image of a set under a function. I have seen several set theory text books that give this impression, but I suspect that their authors have no idea of the logical strength of Replacement. Of course, when you recognise that what mathematical objects DO is what matters, and not what they ARE, you appreciate that the image is simply a quotient by an equivalence relation. I'm sure that set theorists must have realised a long time ago that this could be done within Zermelo set theory. However, no sooner have you rejected that answer than they come up with another, equally fatuous one: Replacement is needed to construct the ordinal omega+omega. Whilst this may actually be true of the kind of intuitionistic ordinals that I called "plump", it is nonsense in a classical setting: a model of omega+omega is provided by the even numbers followed by the odd ones. It's only the set theorists' instistance on representing everything by means of epsilons and nested curly brackets that necessitates Replacement. Giving up on set theorists for a sensible answer, let's return to ordinary mathematics. The usual situation in which Replacement is needed is when you want to form the limit or colimit of some iterated construction. Even then it may be possible to carve the result out from a sufficiently big object that is definable in Zermelo set theory, or in an elementary topos with natural numbers. But there are clear examples in which this cannot be done. One is N-fold iteration of the powerset, starting from N itself, since such a union would itself be a model of Zermelo set theory. An example of this arises in domain theory. Start with the domain X_0 that consists of N+3 points arranged something like this: 0 1 2 3 .... xxxxxxxxxx a b \ / bottom where "xxxxxxxxxx" means that every numeral is above both a and b. Then form the exponentials X_n+1 = X_n ^ X_n. It can be shown that these are algebraic L-domains (in Achim Jung's terminology) whose bases increase in cardinality in the same way as the iterated powersets do. With these examples in mind, we can return to Mike Shulman's original question, or at least some simple example of it: Can we express the transfinite iteration of a functor in purely categorical language? Whether this expresses the full strength of Replacement, I do not know - probably not. However, as I have said, it is impossible to engage set theorists in an intelligent discussion about this, as they seem to be incapable of thinking outside their idiosyncratic representation of mathematical objects as epsilons and curly brackets. Before considering the iteration of functors, we need to have some categorical notion of "ordinal" and "transfinite". In anticipation of anyone else who may feel the need to point this out, let me say first that Andre' Joyal and Ieke Moerdijk studied ordinals as part of the original book on Algebraic Set Theory. However, AST is a "sets and classes" formulation. I too studied (intuitionistic and categorical) ordinals at the same time (the mid 1990s), but in an intrinsic (no classes) way. My first account of this, @article{TaylorP:intso, author = {Taylor, Paul}, title = {Intuitionistic Sets and Ordinals}, journal = {Journal of Symbolic Logic}, volume = 61, year = 1996, pages = {705--744}} developed set theory as (well founded, extensional) epsilon-structure on a carrier, in the same way as we develop group theory as algebraic structure on a carrier. It proved Mostowski's theorem - that any well founded structure has an extensional quotient - WITHOUT using Replacement. The set theory textbooks say that this is necessary, because of their epsilontic representation. It then developed set theory a la Zermelo, and various kinds of intuitionistic ordinals, providing a parallel approach to that in AST, with analogous results. (Intuitionistically, there are several different kinds of ordinals.) Later I studied "well founded coalegebras" and showed that this notion of INDUCTION implies RECURSION, in the form of coalgebra-to-algebra homomorphisms. This is in Section 6.3 of my book, and also in an unpublished paper that you can find on my web site at www.PaulTaylor.EU/ordinals Two years ago, a certain categorist tried to pass this work off as his own, at the Nova Scotia category theory meeting and in the proceedings of "Computer Science Logic". In doing so, he also demonstrated his ignorance of both set theory and category theory as foundations. I hereby warn him that, should he utter a single word on this subject in my hearing, I will identify him and publish full details of the events. My work on this topic was based on a different paper by Gerhard Osius from the one that Colin cited, @article{OsiusG:catstc, author = {Osius, Gerhard}, title = {Categorical Set Theory: a Characterisation of the Category of Sets}, journal = {Journal of Pure and Applied Algebra}, volume = 4, year = 1974, pages = {79--119}, review = {MR 51/643}} which is also summarised in Section 9.2 of "Topos Theory" by Peter Johnstone. It has also been taken up by Capretta, Uustalu and Vene, but in the context of functional programming, not set theory. If there are more recent developments based on the tradition of Osius then I would be pleased to hear about them. Briefly, we express an ordinal number alpha as a coalgebra parse: alpha ---> D(alpha) in the category of posets, where D is the covariant functor that yields the poset of lower subsets. The reason for the name "parse" is given in my book; in this case, its set-theoretic translation is parse (alpha) = { beta | beta epsilon alpha } Now we are in a position to define transfinite iteration of a functor T. This has to be INDEXED (or FIBRED if you prefer), ie it is applied, not to a single object of the category, but to an alpha-indexed family of objects. The display map X--->alpha whose beta'th fibre is the beta-fold iteration of T on the empty set is characterised by the fact that it forms a pullback [beta:alpha, X[beta]] ---> [U:D(alpha), colim {TX[gamma] | gamma in U] | | | | | | |------+ | | | | | | | V parse V alpha ------------------------------> D(alpha) Considered in the category of all discrete fibrations over posets, those for which this is a pullback form reflective subcategory. More precisely, this subcategory is closed under limits, but it only has a reflector if the axiom scheme of replacement holds (say, with the covariant powerset for T). I don't actually think that the ordinals (ie totally ordered sets) as such are needed in this approach. I think one could present it using proofs (of well formedness of terms in a type theory) instead. Then each type theory has a version of Replacement that proves that the given theory is consistent. I got a brief and tiny glimpse of the awesome power of the axiom scheme of replacement in the later stages of writing my book, but that was ten years ago. It appeared to say something like this: if you have some theory that is provably consistent, ie it has a free internal model, then it has a model that is a submodel of the universe in which you are working. Or, "any script can be played out in real life". Please forgive me if I cannot now say this very clearly - I found the idea rather frightening at the time, and it has long since been "swapped out" of my skull. Replacement was conceived by Dimitri Mirimanoff in 1917, and incorporated into Zermelo's axioms by Abraham Fraenkel in 1922. This was well after the fuss over Russell's paradox (1900) had died down, but well before Godel and Turing provided a new generation of paradoxes in the 1930s. So it may be that set theorists had let their guard down in the intervening decades. On Good Friday 1998, I (thought that I had) found a proof that ZF is inconsistent, but the system was resurrected on the third day. I published this "proof" as the final exercise of my book and on "categories" on 1 April 1999; this posting has since being doing the rounds, and I got a reply a couple of years ago. I am certainly not going to spend my time looking for contradictions in set theory, but I find it amazing that anyone should be so confident that Replacement is safe to work with. I also think that it would do Mathematics a great deal of good if someone were to find an inconsistency in ZFC. Paul Taylor From rrosebru@mta.ca Fri Mar 14 20:25:54 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 14 Mar 2008 20:25:54 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JaJ7z-0007Dn-HE for categories-list@mta.ca; Fri, 14 Mar 2008 20:15:55 -0300 Content-Disposition: inline Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset="us-ascii" MIME-Version: 1.0 From: To: categories@mta.ca Subject: categories: Re: categorical formulations of Replacement Date: Fri, 14 Mar 2008 11:56:19 -0400 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 72 The uncountability argument mentioned by Thomas does not apply because Colin's Replacement axiom does not say that "all" external families are representable, only those definable by formulas. Of course the statement is relatively simple only when 1 separates, but that is the case where much of the interest=20 in such axioms has concentrated: categories of pure Cantorian cardinals. Speculating about whether such principles yield anything for the more cohesive sets encountered in geometry and analysis, it develops that, although specifying the internal families is rather easy, explaining which formulas define the appropriate external families is not because of the=20 need to include functorality, sheaf=20 condition,etc. Concerning the 70s work of Cole,Osius,etal, they certainly constructed in lectures an equivalence between the categories of models of bZ and aETCS. Was it not published ?=20 (I don't recall any example showing=20 that the augmentation a was actually needed.) The statement that Replacement is weak seems to follow from using that word to refer to classes derived from an already=20 representable V, rather than the traditional meaning used by Colin, referring to arbitrary formulas. Bill On Thu Mar 13 22:34 , Thomas Streicher sent: >The Replacemnt axiom which Colin formulated in his article in Phil.Math. >only works for well-pointed categories. But even in this framework it is >too strong due to its requirement that every external family arises from an >internal one. So it fails for example for the model of ETCS arising from >a countable model of ZFC because there are only countably many internal >families over N whereas there uncountable many external families indexed >by (the global elements of) N. >A defect of the work from the 70ies (Cole, Osius at.al.) is that it just p= roves >equiconsistency of ETCS and bZ (bounded Zermelo set theory) and not an >equivalence between models of ETCS and bZ. > >I think that the more interesting question is what is a model of intuition= istic >set theoy which cannot be well-pointed. For this purpose it is INDISPENSIB= LE >to have in our category an object U of all sets. >This was first recognized and formulated by Christian Maurer in his paper >"Universes in Topoi" which appeared in the SLNM volume "Model theory and t= opoi" >(ed. Lawvere, Maurer, Wraith). Maurer was working in a topos and postulate= d an >object U (a "universe") of this topos with ext : U >-> P(U) satisfying a f= ew >axioms which ensure that U is a model for IZF (without saying so). >In particular, he has a clear formulation of the axiom of replacement, nam= ely > > (\forall a : U) (\forall f : U^{ext(a)}) (\exists b : U) > > (\forall y : U) (y \in ext(b) (\exists x \in ext(a)) y =3D f(x)) > >albeit in a somewhat less readable since he avoids the internal language >of the topos. > >This point of view was taken up later in the Algebraic Set Theory (AST) of >Joyal and Moerdijk whose work concentrated on CONTRUCTING (initial) univer= ses >of this kind. A couple of years later Alex Simpson in his LiCS'99 paper to= ok >up Maurer's early insight (at least he refers to Maurer's paper) but weake= ned >the ambient category to be a model for first order logic (as set theorists= do). >The main new ingredient of AST (and Alex's paper) is the assumption of a c= lass >of small maps giving (cum grano salis) a notion of "size" (like B'enabou's >calibrations but satisfying much stronger axioms) together with a notion of >small powerset functor P_s (depending on the class of small maps). A unive= rse >is then defined as a(n initial) fixpoint U of P_s which, of course, can't = be >small itself. > >A point which seems to have been overlooked in this latest discussion is >that Replacement per set is not very strong. It gets its usual strength on= ly >in presence of unbounded separation. See the paper by Awodey, Butz, Simpson >and me which appeared in last year's Bull.Symb.Logic (see also >http://homepages.inf.ed.ac.uk/als/Research/Sources/set-models-announce.pdf) >where we discuss this in more detail. The point is that around every topos >EE one can build a category of classes whose small "set" part is equivalent >to EE. The corresponding class theory BIST is thus conservative over topos >logic (with nno). > >Later on Awodey and his students have also studied the much weaker "predic= ative" >case where replacement still holds. See also Aczel and Rathjen's work on C= ZF >in this context which is much older than AST dating back to articles by Ac= zel >in the late 70ies (and based on previous work by John Myhill). > >Thomas Streicher > > > From rrosebru@mta.ca Fri Mar 14 20:25:54 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 14 Mar 2008 20:25:54 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JaJ6N-00076a-K0 for categories-list@mta.ca; Fri, 14 Mar 2008 20:14:15 -0300 From: Colin McLarty To: categories@mta.ca Date: Fri, 14 Mar 2008 09:52:28 -0400 MIME-Version: 1.0 Content-Language: en Subject: categories: Re: categorical formulations of Replacement X-Accept-Language: en Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 73 Thomas Streicher Friday, March 14, 2008 7:55 am writes > The Replacement axiom which Colin formulated in his article in > Phil.Math.only works for well-pointed categories. But even in this > framework it is > too strong due to its requirement that every external family arises > from an internal one. What do you mean by an "external family"? Do you mean every family that the mathematician looking at the model from outside it would recognize, or every family defined by a relation in the first-order language? Are you just invoking the Skolem paradox in a categorical setting? What is the axiom scheme "too strong" to do? > A defect of the work from the 70ies (Cole, Osius at.al.) is that it > just proves > equiconsistency of ETCS and bZ (bounded Zermelo set theory) and not an > equivalence between models of ETCS and bZ. Before I can comment I have to ask, do you mean equivalence between models, or between the categories of models? Exactly what is it that you want but feel that work does not prove? Is it just that you prefer to think about a different question? As you put it: > I think that the more interesting question is what is a model of > intuitionistic set theory which cannot be well-pointed. For this > purpose it is INDISPENSIBLE to have in our category an object U > of all sets. You must use these words differently than I do. We normally say every topos is a model of intuitionistic set theory. Many are not well-pointed yet have no object of all sets. Synthetic Differential Geometry (in the full, topos version, not Synthetic Infinitesimal Analysis as in Bell's book) is one of the best-known axiomatic extensions of the elementary topos axioms. It has no well-pointed models yet its usual models have no object U of all sets. > A point which seems to have been overlooked in this latest > discussion is > that Replacement per set is not very strong. It gets its usual > strength only in presence of unbounded separation. That is in an intuitionistic setting. In classical logic, unbounded replacement implies unbounded separation. The "replacing set theory" thread was about replacing ZFC, which has classical logic. You cite the very nice work you have done Steve Awodey, Carsten Buts, and Alex Simpson. But that work does not aim (just) at axiomatizing the classical universe of sets. There are different questions here. best, Colin From rrosebru@mta.ca Fri Mar 14 20:25:55 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 14 Mar 2008 20:25:55 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JaJ8y-0007HE-To for categories-list@mta.ca; Fri, 14 Mar 2008 20:16:56 -0300 Date: Fri, 14 Mar 2008 11:02:39 -0500 From: "Michael Shulman" Subject: categories: Re: categorical formulations of Replacement To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 74 On Thu, Mar 13, 2008 at 9:34 PM, Thomas Streicher wrote: > But even in this framework it is > too strong due to its requirement that every external family arises from an > internal one. So it fails for example for the model of ETCS arising from > a countable model of ZFC because there are only countably many internal > families over N whereas there uncountable many external families indexed > by (the global elements of) N. I do not understand what is meant by "external" here. What Colin and Osius's replacement scheme states is that every *definable* family of sets is internal. This is the same as the ZF replacement axiom: every *definable* function defined on a set is a set. Since there are only countably many logical formulas, there are only countably many definable families for them to define, so there is no problem with countable models. > A defect of the work from the 70ies (Cole, Osius at.al.) is that it just proves > equiconsistency of ETCS and bZ (bounded Zermelo set theory) and not an > equivalence between models of ETCS and bZ. In Osius' paper "Categorical set theory" he does prove exactly this sort of equivalence, by adding a couple weaker extra axioms to ETCS and bZ relating to the existence of transitive closures and collapses. An account can also be found in Johnstone's "Topos Theory", chapter 9. Mike From rrosebru@mta.ca Fri Mar 14 20:25:55 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 14 Mar 2008 20:25:55 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JaJ9u-0007LD-Jh for categories-list@mta.ca; Fri, 14 Mar 2008 20:17:54 -0300 From: Thomas Streicher Subject: categories: I was partly wrong To: categories@mta.ca Date: Fri, 14 Mar 2008 19:26:39 +0100 (CET) MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 75 It is, of course, true that Maurer's and AST's form of replacement is stronger than the SCHEMA of replacement because it quantifies over all families in the topos and not just the syntactically definable ones. The same "mistake" appears in the definition of Grothendieck universes. Set theorist insist on formulating set theory in first order logic and therefore must formulate replacement as a schema since they can't quantify over families of sets (the ambient category is a logos or Heyting category but not cartesian closed). If they decided to formulate set theory within higher order logic they presumably would have formulated replacement not as a schema (see Bill's remarks on Goedel and Kreisel's suggestions). But of course there is no problem to adapt Maurer's notion of replacement to the schema form. Nevertheless, since in the schema of replacement there may occur free variables one has to be careful when quantifying over functions representable by formulas in the language of set by considering all reindexings along U^n -> 1. This is not done in McLarty's article. He just gives a reformulation of the replacement schema WITHOUT free variables. I presumably tried to pack too much into my mail and thus (again) forgot about the distiction between schema and the stronger from (sorry Bill!). My intention was to advocate the necessity of an object U of sets which is not available in (models of) ETCS. So one cannot directly interpret the language of set theory in a model of ETCS but has first to CONSTRUCT a model for set theory from it. But this is only possible because models of ETCS are wellpointed. So if one is interested in models of constructive set theory ETCS is not particularly helpful. Thomas From rrosebru@mta.ca Fri Mar 14 20:25:55 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 14 Mar 2008 20:25:55 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JaJ6q-00078d-HG for categories-list@mta.ca; Fri, 14 Mar 2008 20:14:44 -0300 From: Thomas Streicher Subject: categories: old stuff available on net To: categories@mta.ca Date: Fri, 14 Mar 2008 15:26:13 +0100 (CET) MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 76 Now and then I receive a request for my Habilitation Thesis from 1993 on Semantics of Intensional Type Theory which was never published and is not available in one of the usual electronic formats. But if you really want to look at it you can find a scanned version at www.mathematik.tu-darmstadt.de/~streicher/HabilStreicher.pdf BUT it's awfully big (4915468 KB). Thomas Streicher From rrosebru@mta.ca Sat Mar 15 10:05:51 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 15 Mar 2008 10:05:51 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JaVzr-00030A-86 for categories-list@mta.ca; Sat, 15 Mar 2008 10:00:23 -0300 Date: Fri, 14 Mar 2008 23:49:49 -0500 From: "Michael Shulman" Subject: categories: Re: I was partly wrong To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 77 On Fri, Mar 14, 2008 at 1:26 PM, Thomas Streicher wrote: > Set theorists insist on formulating set theory in first order logic > and therefore must formulate replacement as a schema since they > can't quantify over families of sets (the ambient category is a > logos or Heyting category but not cartesian closed). If they > decided to formulate set theory within higher order logic they > presumably would have formulated replacement not as a schema (see > Bill's remarks on Goedel and Kreisel's suggestions). I have read that Zermelo always insisted on a second-order interpretation of his axioms; in particular as regards his notion of a "definite" property. It was only later mathematicians who decided that "definite" should mean "first-order". Also, the perfectly classical von Neumann-Bernays-Godel set theory, which set theorists know about and make use of, does use classes to avoid axiom schemas in the same way that AST does, thereby obtaining a finitely axiomatizable theory equivalent to ZFC. Having said that, I will now dive off the deep end into what seems likely to be very hot water (-:, and say that it seems to me that "higher-order logic" is really first-order set theory in disguise. The interpretation of "higher-order quantifiers" ranging over "sets of blah objects" (or "predicates" or "classes") only makes sense with respect to an external set theory that specifies what counts as a "set"---or by introducing a new type called "set of blah objects", which brings you back into to a first-order set theory. I think this is more evident now than it was to Zermelo, now that we have, say, the independence of CH before our eyes to convince us that the extension of the notion of "subset of a given set" is not uniquely determined by the idea of "set"---at least, not obviously---and requires some sort of theory of sets to specify what it means. I think that this is one reason that modern set theorists usually stick to first-order logic. This is not to dispute the value and power of what is normally called second-order and higher-order logic. In particular, what is called higher-order logic is essential to topos theory and AST, for which I have a great appreciation and respect. But I think it does show that if you get down to the root, all mathematics is actually done in first-order logic. For instance, the "higher-order" formulation of replacement that you advocate is still part of a *first-order* theory of sets and classes (AST). Now I'm not saying there's anything wrong with that! It certainly gives rise to interesting, powerful, and useful mathematics, which I don't understand nearly as well as I would like to. But I don't think it is necessarily indicative of blindness on the part of set theorists that they choose to include only sets in their first-order logic. There is something about introducing classes as first-class objects that makes some of us slightly uneasy: what is it that distinguishes a set from a class? What exactly prevents us from talking about classes of classes? Why did we draw an arbitrary line at some point and say "any collection of things bigger than *this* is a class", and forbid ourselves from doing otherwise intuitive things to them like taking power objects and exponentials? Why shouldn't set theory be the *universal* theory of "collections of things"? Of course, it is likely that I am quite wrong, so I await correction. (-: Best, Mike From rrosebru@mta.ca Sat Mar 15 19:57:53 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 15 Mar 2008 19:57:53 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JafGS-0004ci-Kz for categories-list@mta.ca; Sat, 15 Mar 2008 19:54:08 -0300 From: Colin McLarty To: categories@mta.ca Date: Sat, 15 Mar 2008 17:15:25 -0400 MIME-Version: 1.0 Content-Language: en Subject: categories: Re: replacing set theory Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 78 wlawvere@buffalo.edu Thursday, March 13, 2008 9:20 pm Wrote, with much else, about an equivalent to the axiom scheme of replacement in ETCS, and his mimeographed notes on it now reprinted at TAC http://138.73.27.39/tac/reprints/articles/11/tr11abs.html > Using the definable > classes of sets smaller than any > given set, the postulate that there > are arbitrarily large such classes > closed under arbitrary definable > operations phi is proposed. I am > not aware of any further studies > of that postulate. It is right at the end, on p. 34 of the reprint. To me, it is really a very nice stream-lined version of a reflection principle and deserves more of a look. Up to now I have tended to translate it into a replacement scheme because they are more familiar to my audience. But it is much more elegant as it cuts straight down to a very elegant isomorphism-invariant principle. best, Colin From rrosebru@mta.ca Sat Mar 15 19:57:54 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 15 Mar 2008 19:57:54 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JafFD-0004ZL-UD for categories-list@mta.ca; Sat, 15 Mar 2008 19:52:52 -0300 From: Thomas Streicher Subject: categories: internal versus external To: categories@mta.ca Date: Sat, 15 Mar 2008 17:31:54 +0100 (CET) MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 79 I try to untangle various threads in the recent discussion. (1) Yes, there is a difference between replacement as a schema and the stronger form of replacement as formulated in Groth. universes. (To my embarrasment I have blurred this distinction against better knowledge!) (2) One has to carefully distinguish between the following two questions (a) find as good as possible an axiomatization of "the" category of sets which was the aim of ETCS and Cole, Mitchell and Osius; in this situation it is natural to assume wellpointedness (b) to ask what are good notions of model for intuitionistic set theories where the assumption of wellpointedness is, of course, much too restrictive. (a) and (b) are so different questions that I wouldn't go for comparing them. Reading Mike Shulman's original mail I had the impression he was going for (b) when asking whether certain arguments involving transfinite recursion could be done categorically. There the answer is in general not. Alas, I can't say anything about the most interesting question about small objects argument. But I know the answer in case of the classical example where replacement is needed, namely Borel determinacy. Don Martin gave a proof of Borel determinacy iterating the powerset functor \omega_1 times and later H.Friedman showed that axiom of replacement is indispensible for proving Borel determinacy. One also knows that in the realizability model of IZF (Friedman, McCarty et.al) Borel determinacy fails although it validates replacement (which is part of the axioms of IZF). Now I am coming to the real issue which triggered me to take part in the discussion, namely an apparent confusion between internal and external families of objects. I thought that on pp.77-79 of my lecture notes on fibrations (www.mathematik.tu-darmstadt.de/~streicher/FIBR/FibLec.pdf.gz) I gave a summary of the "current view" on these questions. But in the light of the recent discussion I think a few more words seem to be in place. Suppose EE is a topos. The most immediate notion of external family of objects is a function I -> Ob(EE) where I is a set. An internal family in EE is simply a map A -> X in EE. Obviously, these gadgets are quite different because in general sets aren't objects of EE and objects of EE aren't sets. However, one can turn an internal family a : A -> X into an external family of objects in EE indexed by EE(1,X), namely by associating with x : 1 -> X the object x^*a, i.e. the (source of the) pullback of a along x. Now one may ask whether this is a 1-1-correspondence for general toposes EE. There is a simple example showing that going from internal to external one looses information: let GG be a (finite) nontrivial group and G the representable object of Psh(GG) then ALL morphisms to G (and there are plenty) represent the external empty family of objects of Psh(GG). This might lead one to think that internal families are just "intensional representations" of external families. But this view is mistaken since there are external families of objects of EE which don't arise by externalising internal families. This is obvious since if EE is a small topos (say the free topos with nno) then there are I-indexed families of EE where I exceeds the cardinality of any EE(1,X). But there is the much more refined example of Peter Johnstone (also described in my notes on the web) of a topos EE over Set and an NN-indexed family of objects (A_n) in EE which doesn't arise from an internal family in EE. The topos EE is the full subcat of actions of the group (ZZ,+) on those objects where there is a finite bound on the size of orbits and A_n is ZZ_n acted on by ZZ in the obvious way. The reason why this family cannot arise from an internal family a : A -> X in EE is that there is a finite bound on the size of all orbits in A and thus on all orbits in the A_x (x \in EE(1,X). But of course A_n has an orbit of length n and, accordingly, there is not a finite bound on the orbits of all the A_n. Of course, if EE is a wellpointed topos the situation is "better". If EE has also small sums then we clearly have a 1-1-correspondence between internal families indexed by X and and external families indexed by EE(1,X). But this isn'the case anymore if we drop the assumption of small sums. Let EE be the wellpointed small topos arising from a countable model of Z (Zermelo's set theory, i.e. ZF without replacement schema). Then for reasons of cardinality there are EE(1,N)-indexed external families which don't arise from an internal one. >From some of the mails I got the impression that it is claimed that restriction to "syntactically definable" (s.d.) families allows one to identify internal and external. This is definitely wrong for the non-wellpointed case due to the many different representations of the empty family. It might be the case that any "syntactically definable" external family indexed by EE(1,X) arises from some internal family a : A -> X though I don't see how to prove it. (Does any of the advocates of external families have a proof for that?) Finally, I would like to point out that there are two different meanings of the phrase "syntactically definable" family. The one used by Colin in his Philosophia article means "syntactically definable in the language of ETCS" and the one used by set theorists is "definable in the language of set theory". The latter hasn't any meaning in a model for ETCS since such a model doesn't allow one to interpret a formula in the language of set theory. But, of course, one can interpret set theoretic formulas in the models of bZ constructed from a model of ETCS. The reason why I doubt that models of ETCS and bZ are equivalent is that when going from a model of ECTS to a model of bZ is that one has to restrict to the well-founded part. At least that's what I recollect from MacLane and Moerdijk's exposition in their book. But I am ready to believe that adding wellfoundedness axioms to ETCS can remedy this situation. Of course, I might be all wrong with what I have said above. But then I would like to know what is the appropriate notion of "(external) family of objects" that one should use in case of non-wellpointed toposes. Thomas PS For sake of completeness I can't refrain from mentioning the following considerations I was told by Jean B'enabou some years ago. Let EE be some topos (J.B. was of course considering more general situations). Then split fibrations over EE correspond to categories internal to Psh(EE) = Set^{EE^op} (where Set is chosen big enough for EE being small w.r.t. Set). Then one my consider the presheaf set(EE) where set(EE)(X) consists of representable presheaves over EE/X. It is natural to define an X-indexed family of objects of EE as a morphism y(X) -> set(EE) but by Yoneda this is an element of set(EE)(X) corresponding to a map to X in EE. From rrosebru@mta.ca Sun Mar 16 10:16:28 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 16 Mar 2008 10:16:28 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JasUj-00016W-3s for categories-list@mta.ca; Sun, 16 Mar 2008 10:01:45 -0300 Date: Sun, 16 Mar 2008 04:32:37 +0300 From: Jawad Abuhlail Subject: categories: The Category of Semimodules over Semirings To: categories@mta.ca MIME-version: 1.0 Content-type: text/plain; charset=us-ascii Content-transfer-encoding: 7BIT Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 80 Dear colleagues, A semiring is roughly speaking a ring without subtraction, i.e. (R,+,0) is an Abelian monoid & (R,*,1) is a semigroup with distribution of * over + (e.g. the set of non-negative integers). A semimodule over a semiring is roughly speaking a module without subtraction, i.e. (M,+,0) is an Abelian Monoid, and there is a scalar multiplication of the semiring on M with the usual expected properties. A semimodule over a semiring is cancellative, if the Abelian monoid (R,+) is cancellative. The category of N0-Semimoduels is just the category of "Commutative Monoids". Indeed the category of left semimodules over an arbitrary semiring R? (A special example would be the category of commutative monoids) is not pre-additive. However, for any left semimodules M and N over a semiring R, (Hom_R(M,N),+,0) is an Abelian monoid and it has kernels and cokernels. A monomorphism of semimodules is injective, however only an "image-regular" epimorphism is subjective. For a morphism of semimodules f: M --> N, the sequence 0 ---> Coimage(f) --> Im(f) --> is exact, but the canonical map Coimage(f) --> Im(f) is not an isomorphism (neither a bimorphism), unless f is regular. The category of semimodules had products, equalizers and products (however not necessarily coequalizers). The category of cancellative semimodules is complete and cocomplete. It has a generator, namely the semiring itself and I also "expect" it to have exact colimits (ANY REFERENCE?). What kind of Categories is the category of (cancellative) semimodules over semiring? Is there a notion of "almost Grothendieck categories" or "Semi-Grothendieck Categories" to which categories of (cancellative) semimodules fit? Unfortunately, I did not find a single book that clarifies the categorical aspects of semimodules (The 3 books of Golan as well as the books on the subject by others are devoted more to semirings and automata and not have much on semimodules). I appreciate very much your comments, suggestions for literature (e.g. books, Ph.D. dissertations, articles) on the CATEGROY OF (Cancellative) SEMIMODULES (other than the papers of Takahashi and Katsov, which I already have). With best regards, Jawad ----------------------------------------------------- Dr. Jawad Abuhlail Dept. of Math. & Stat. Box # 5046, KFUPM 31261 Dhahran (KSA) http://faculty.kfupm.edu.sa/math/abuhlail From rrosebru@mta.ca Sun Mar 16 21:56:29 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 16 Mar 2008 21:56:29 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Jb3Zi-0005r8-6V for categories-list@mta.ca; Sun, 16 Mar 2008 21:51:38 -0300 Mime-Version: 1.0 (Apple Message framework v624) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: <844283ef33e889f4665922a47f56511d@PaulTaylor.EU> Content-Transfer-Encoding: 7bit From: Paul Taylor Subject: categories: replacement and iterated powersets Date: Sun, 16 Mar 2008 16:25:31 +0000 To: Categories list Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 81 Maybe I should wave my hands a bit less, and actually spell out a concrete example. Here is how you can say that the display map X-->Nx2 in an elementary topos with a natural numbers object is the sequence of iterated powersets, starting with X[0,0]=N, where X[0,1] is the union of X[n,0]. In set-theoretic language, X is then the cardinal beth_{omega.2}. Of course, the following is not a CONSTRUCTION of this display map, just a SPECIFICATION of it: we need Replacement to say that there EXISTS a display map that satisfies this specification. First we define the strict arithmetical order on Nx2: (n,0) < (m,0) if n < m (n,0) < (m,1) always (n,1) < (m,1) if n < m (n,1) < (m,0) never Nx2 is also a poset, with the reflexive order <= defined as < or =. Let D(Nx2) be the lattice of lower subsets of Nx2 wrt <=. Let parse: Nx2 --> D(Nx2) by parse(p) = {q | q < p}. Now let X-->Nx2 be a discrete fibration in Pos. This means that, whenever q<=p, there is a function X[q]->X[p], and this system respects identities and composition. The powerset functor of the ambient elementary topos is fibred, so we can construct another discrete fibration Y-->D(Nx2) in which Y[U] = colim {Y[q] | q in U}. In particular, if U = parse (0,1) = { (n,0) | n in N }, Y[U] is the colimit of Y[n,0] for n in N and the maps between them. Now, we need to say that Y[parse(p)] = X[p], up to isomorphism. But this is just the statement that X ---------> Y | | | | | | |----+ | | | | | V parse V Nx2 -------> D(Nx2) is a pullback. Paul Taylor From rrosebru@mta.ca Sun Mar 16 21:56:30 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 16 Mar 2008 21:56:30 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Jb3ZG-0005p9-R6 for categories-list@mta.ca; Sun, 16 Mar 2008 21:51:10 -0300 Mime-Version: 1.0 (Apple Message framework v624) Content-Type: text/plain; charset=US-ASCII; format=flowed Content-Transfer-Encoding: 7bit From: Paul Taylor Subject: categories: replacement and the gluing construction Date: Sun, 16 Mar 2008 15:46:42 +0000 To: Categories list Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 82 The words "internal", "external" and "definable" has been used in this discussion of the axiom scheme of replacement, but I believe that we have to be very careful in using them. So far as I can gather, Thomas Streicher is still discussing the "sets and classes" view of replacement. At any rate, I cannot see what he might mean categorically by saying that > an external family of objects is a function I -> Ob(EE) where I is a set unless his Ob(EE) is a "class" in either the sense of algebraic set theory (ie an object in another category besides EE) or some similar approach. AST is an intellectually valid point of view on which much good category theory has been done in the past few years. However, Mike Shulman and I would both like to know how one might formulate replacement WITHOUT classes. If I have misunderstood you, Thomas, then I apologise, but this is an important and difficult topic, so more and clearer explanation is always valuable. If I, who have thought about this myself, do not follow you, then nor do many of the other readers of this forum. Before explaining why I think the word "definable" is also dangerous, let me repeat something about the difference between set theory and category theory. In a word, they say that they are describing mathematical objects up to equality, whilst we say that this can only be done up to isomorphism. However, in the context of replacement in particular, this means that we may end up talking completely at cross purposes. As I said before, they claim that replacement is necessary to construct the ordinal omega+omega, whilst this order structure can be constructed up to isomorphism very easily without it. But consider the example of the N-indexed sequence of iterated powersets of N. The structure that Replacement yields is, up to isomorphism, just N itself, but the force of the axiom-scheme in set theory is that the elements of this set are the iterated powersets. I am not inclined to believe that Cantor understood the strength of replacement when he stated in 1899 that "two equivalent [ie bijective] multiplicities are either both sets or both equalivalent", because his mathematical beliefs were an extreme form of Platonism, and the appreciation of the strength of logic theories simply did not exist at the time. Also, Fraenkel made a mistake in his first attempt to formulate Replacement, saying something that was already derivable from Zermelo's set theory. I believe that, in order to grasp the meaning of Replacement and then formulate it in categorical language, we first have to have a clear understanding of the ways in which we make "definitions" and then implement them. The distinction between planning and execution is very clear if you're building a house, but often very difficult to make in mathematical research. Instead of logic, let's consider algebra. You can define the notion of group using (1) just the symbols for identity and binary multiplication. Or, you can have (2) multiplications symbols of all arities, but still with a single result; this is called the "clone". Next, you can form (3) families of products; this is the "Lawvere theory". There are also (4) the category of all groups and (5) the classifying topos. Of these, (1) is finite, (2,3) require the natural numbers or an axiom of infinity, and (4,5) require a set theory. In some sense, they all capture the same algebraic notion, but foundationally, something more is needed to prove their equivalence. Thomas Streicher uses indexed/fibred category theory to link internal and external notions. I don't see, however, how he manages to give the SPECIFICATION that a display map X-->N is the sequence of iterates of the powerset on N. The formulation that I gave at the end of my previous posting and in Remark 9.6.16 in my book does do this. Earlier in this thread, Paul Levy said, > I'd also like to suggest that "foundations" is being used in > two very different senses. In FoM, it's about quantifying the > philosophical risks involved in particular formal systems and > proofs, i.e. issues such as relative consistency, omega-consistency, > etc. For this purpose the primacy of membership vs composition is > quite immaterial. One could, I suppose, make a formal theory based > on composition equal in strength (in whatever sense) to ZF. There are two important ideas in category theory that have something to say about consistency: (1) Andre' Joyal's categorical formulation of Godel's INCOMPLETENESS theorem; this compares - PROVABILITY, in the form of the internal term or free model of the system (an arithmetic universe, but it could equally be an elementary topos) with - TRUTH in the world within which this is constructed; and (2) the Artin gluing / Freyd cover / scone (Sierpinski cone) / logical relations construction, which can be used to prove CONSISTENCY of various theories, once again by comparing the term model with the ambient universe. Whilst writing a book is career suicide, it does have the intellectual value of forcing one to reconcile apparently conflicting results like these. (See sections 7.7 and 9.6 of my book.) The gluing construction works because it uses the axiom-scheme of replacement. Thomas Streicher and Thorsten Altenkirch, who have contributed to the present thread, have written papers about this construction and its applications. Did they or anybody else draw attention to the need for Replacement in their work on this construction? Categorically, the gluing construction is a comma category that, in the applications to consistency, combines the "semantic" category (eg Set, with Replacement) with the "syntactic" one (the free one, constructed from the types, terms and proofs of the theory under consideration). Roughly speaking, the comma category inherits any structure that both of the original categories had, and the projection to the syntactic category preserves it. Since the syntactic category is the free one, it therefoe has a unique (up to unique iso) structure-preserving functor to the comma category. Combined with the projection to the semantic category, this yields an internal model in which the type constructors (powerset, for example) of the theory are interpreted as the actual semantic powersets or whatever. Steve Awodey, Bill Lawvere, Colin McLarty and Thomas Streicher have presented various views on Replacement. In order that the rest of us might get a better understanding of what each of them means, and how their points of view are related, I invite them each to give an explanation of this foundational aspect of the gluing construction, as they would formulate it according to their respective points of view. From rrosebru@mta.ca Sun Mar 16 21:56:30 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 16 Mar 2008 21:56:30 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Jb3Yh-0005m7-4z for categories-list@mta.ca; Sun, 16 Mar 2008 21:50:35 -0300 From: "Katsov, Yefim" To: Date: Sun, 16 Mar 2008 10:49:40 -0400 Subject: categories: RE: The Category of Semimodules over Semirings Accept-Language: en-US Content-Language: en-US Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: quoted-printable MIME-Version: 1.0 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 83 Dear Jawad, Here are some suggestions that hopefully may help you: 1) In regard of literature on the subject, I'd suggest to look at the book,= "A Guide to the Literature on Semirings and their Applications in Mathemat= ics and Information Sciences (With Complete Bibliography)," by Kazimierz Gl= azek, Kluwer Academic Publishers, 2002. 2) Concerning categorical aspects of semimodules categories, I'd suggest to= check publications of Ildar S. Safuanov whose Ph.D. dissertation, supervis= ed by late Prof. L.A. Skornyakov (Moscow State U., MGU) in 80-th, was about= categorical aspect of semimodules. With my best wishes, Yefim ____________________________________________________________________ Prof. Yefim Katsov Department of Mathematics & CS Hanover College Hanover, IN 47243-0890, USA telephones: office (812) 866-6119; home (812) 866-4312; fax (812) 866-7229 -----Original Message----- From: cat-dist@mta.ca [mailto:cat-dist@mta.ca] On Behalf Of Jawad Abuhlail Sent: Saturday, March 15, 2008 9:33 PM To: categories@mta.ca Subject: categories: The Category of Semimodules over Semirings Dear colleagues, A semiring is roughly speaking a ring without subtraction, i.e. (R,+,0) is an Abelian monoid & (R,*,1) is a semigroup with distribution of * over + (e.g. the set of non-negative integers). A semimodule over a semiring is roughly speaking a module without subtraction, i.e. (M,+,0) is an Abelian Monoid, and there is a scalar multiplication of the semiring on M with the usual expected properties. A semimodule over a semiring is cancellative, if the Abelian monoid (R,+) is cancellative. The category of N0-Semimoduels is just the category of "Commutative Monoids". Indeed the category of left semimodules over an arbitrary semiring R? (A special example would be the category of commutative monoids) is not pre-additive. However, for any left semimodules M and N over a semiring R, (Hom_R(M,N),+,0) is an Abelian monoid and it has kernels and cokernels. A monomorphism of semimodules is injective, however only an "image-regular" epimorphism is subjective. For a morphism of semimodules f: M --> N, the sequence 0 ---> Coimage(f) --> Im(f) --> is exact, but the canonical map Coimage(f) --> Im(f) is not an isomorphism (neither a bimorphism), unless f is regular. The category of semimodules had products, equalizers and products (however not necessarily coequalizers). The category of cancellative semimodules is complete and cocomplete. It has a generator, namely the semiring itself and I also "expect" it to have exact colimits (ANY REFERENCE?). What kind of Categories is the category of (cancellative) semimodules over semiring? Is there a notion of "almost Grothendieck categories" or "Semi-Grothendieck Categories" to which categories of (cancellative) semimodules fit? Unfortunately, I did not find a single book that clarifies the categorical aspects of semimodules (The 3 books of Golan as well as the books on the subject by others are devoted more to semirings and automata and not have much on semimodules). I appreciate very much your comments, suggestions for literature (e.g. books, Ph.D. dissertations, articles) on the CATEGROY OF (Cancellative) SEMIMODULES (other than the papers of Takahashi and Katsov, which I already have). With best regards, Jawad ----------------------------------------------------- Dr. Jawad Abuhlail Dept. of Math. & Stat. Box # 5046, KFUPM 31261 Dhahran (KSA) http://faculty.kfupm.edu.sa/math/abuhlail From rrosebru@mta.ca Sun Mar 16 21:56:30 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 16 Mar 2008 21:56:30 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Jb3ag-0005vj-8i for categories-list@mta.ca; Sun, 16 Mar 2008 21:52:38 -0300 Date: Sun, 16 Mar 2008 13:43:24 -0400 From: "Fred E.J. Linton" To: Subject: categories: Re: The Category of Semimodules over Semirings Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 84 On Sun, 16 Mar 2008 09:26:35 AM EDT Jawad Abuhlail = wrote, in part, on the Subject: The Category of Semimodules over Semiring= s, > ... The category of semimodules had products, equalizers and > products (however not necessarily coequalizers). = I must be missing something here. Don't the (say, left-) semimodules (over a given semiring) constitute an equationally definable class = of algebras? That is, aren't they determined entirely by operations = and equations? If they DO, that is, if they ARE, then the category of them all (together= with their homomorphisms) must, like all such "varietal categories," have= = all (small) limits and colimits, and, in particular, all coequalizers. Alas, I have little else to offer. Cheers, and Happy St. Paddy's Day, -- Fred From rrosebru@mta.ca Mon Mar 17 09:32:38 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Mar 2008 09:32:38 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JbEP2-0002de-EV for categories-list@mta.ca; Mon, 17 Mar 2008 09:25:20 -0300 MIME-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: quoted-printable Subject: categories: Re: The Category of Semimodules over Semirings Date: Mon, 17 Mar 2008 13:25:27 +1100 From: "Stephen Lack" To: Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 85 As Fred says, the semimodules over a given semiring are=20 determined by operations and equations, and so are complete and cocomplete. In terms of exactness properties they are also=20 (i) locally finitely presentable (so that finite limits commute with filtered colimits, and=20 (ii) Barr-exact (so that there is an equivalence between quotients and congruences) If we restrict to the cancellative case, we still have completeness and cocompleteness and (i), but (ii) fails. =20 Steve Lack. > -----Original Message----- > From: cat-dist@mta.ca [mailto:cat-dist@mta.ca] On Behalf Of=20 > Fred E.J. Linton > Sent: Monday, March 17, 2008 4:43 AM > To: categories@mta.ca > Subject: categories: Re: The Category of Semimodules over Semirings >=20 > On Sun, 16 Mar 2008 09:26:35 AM EDT Jawad Abuhlail=20 > wrote, in part, on the Subject: The=20 > Category of Semimodules over Semirings, >=20 > > ... The category of semimodules had products, equalizers=20 > and products=20 > > (however not necessarily coequalizers). >=20 > I must be missing something here. Don't the (say, left-)=20 > semimodules (over a given semiring) constitute an=20 > equationally definable class of algebras? That is, aren't=20 > they determined entirely by operations and equations? >=20 > If they DO, that is, if they ARE, then the category of them=20 > all (together with their homomorphisms) must, like all such=20 > "varietal categories," have all (small) limits and colimits,=20 > and, in particular, all coequalizers. >=20 > Alas, I have little else to offer. Cheers, and Happy St. Paddy's Day, >=20 > -- Fred >=20 >=20 >=20 >=20 >=20 From rrosebru@mta.ca Mon Mar 17 10:43:31 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Mar 2008 10:43:31 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JbFYC-0005u4-QH for categories-list@mta.ca; Mon, 17 Mar 2008 10:38:52 -0300 Date: Mon, 17 Mar 2008 11:05:53 +0100 From: Joachim Kock Subject: categories: Workshop on Categorical Groups To: categories@mta.ca, algtop-l@lists.lehigh.edu MIME-version: 1.0 Content-type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 86 WORKSHOP ON CATEGORICAL GROUPS June 16 to 20, 2008 Institut de Matem=E0tica de la Universitat de Barcelona This is an event within the CRM thematic year on Homotopy=20 Theory and Higher Categories (http://www.crm.cat/hocat/). The workshop will focus on recent developments in the theory=20 of categorical groups and related topics, as well as their=20 applications to higher-order geometry and theoretical physics. The following have agreed to give keynote talks: - John Baez (University of California at Riverside) - Andr=E9 Joyal (Universit=E9 du Qu=E9bec =E0 Montr=E9al) - Behrang Noohi (Florida State University) - Tim Porter (National University of Ireland) - Enrico Vitale (Universit=E9 Catholique de Louvain) Further information will gradually be made available at http://mat.uab.cat/~kock/crm/hocat/cat-groups/. The deadline for registration is May 31, 2008. The organisers, Pilar Carrasco Josep Elgueta Joachim Kock Antonio Rodr=EDguez Garz=F3n From rrosebru@mta.ca Mon Mar 17 10:43:31 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Mar 2008 10:43:31 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JbFWc-0005Yt-RP for categories-list@mta.ca; Mon, 17 Mar 2008 10:37:15 -0300 Date: Mon, 17 Mar 2008 05:01:47 -0400 From: "Sanjeevi Krishnan" To: categories@mta.ca Subject: categories: First announcement of ATMCS III conference in Paris, France and Call for Papers MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 87 Please accept my apologies for duplicate emails, as this announcement has been sent to multiple mailing lists. I would like to announce the following conference and a first call for papers. ATMCS III Algebraic Topological Methods in Computer Science Paris, France 7-11 July 2008 http://www.lix.polytechnique.fr/~sanjeevi/atmcs/ ***FIRST CALL FOR PAPERS*** Deadline for submission of abstract: 15 May 2008 Notification of acceptance: 5 June 2008 Deadline for registration: 15 June 2008 Conference: 7-11 July 2008 contact information: atmcs08@lix.polytechnique.fr Recent research has shown that techniques from algebraic topology adapt strikingly well in studying computational systems and other subjects within Computer Science. This third ATMCS conference hopes to bring together researchers employing geometric/topological methods in both abstract and concrete areas of computer science. The week long conference will feature some invited talks, several accepted talks, a poster session, and countless opportunities for informal collaboration; we plan to publish our proceedings in a refereed journal, pending approval. All authors submitting an abstract by the deadline will have an opportunity, at the least, to present a (refereed) poster at the poster session. ***SCOPE*** Areas of interest include, but are not limited to, concurrency theory, distributed computing and complexity, rewriting systems, image analysis, and sensor networks. ***INVITED SPEAKERS*** A current (and incomplete) list of plenary speakers for the conference includes: John Baez, University of California Riverside, U.S.A. Gunnar Carlsson, Stanford University, U.S.A. Herbert Edelsbrunner, Duke University, U.S.A. Emmanuel Haucourt, CEA and Ecole Polytechnique, France Rick Jardine, University of Western Ontario, Canada Sanjeevi Krishnan, CEA and Ecole Polytechnique, France Claudia Landi, University of Modena e Reggio Emilia, Italy Francois Metayer, University of Paris 7 and CNRS, France Konstantin Mischaikow, Rutgers University, U.S.A. Gaucher Philippe, University of Paris 7 and CNRS, France Francis Sergeraert, University of Grenoble 1, France ***INSTRUCTIONS FOR SUBMISSIONS*** Authors are invited to submit extended abstracts summarizing current work that explores connections between algebraic topology and computer science. All abstracts should be written in English and should not exceed 1 single-spaced page. Although abstracts preferrably should be sent by email to atmcs08@lix.polytechnique.fr, abstracts may also be mailed to the postal address: Sanjeevi Krishnan DRT LIST DTSI SOL MEASI CEA Saclay 91191 Gif sur Yvette Cedex In all cases, submission materials must arrive by May 15, 2008. ***PROGRAM COMMITTEE*** Gunnar Carlsson, Stanford University, U.S.A. Pierre Louis Curien, CNRS and University of Paris 7, France Massimo Ferri, Bologna University, Italy Eric Goubault, CEA and Ecole Polytechnique, France Maurice Herlihy, Brown University, Providence, U.S.A. Yves Lafont, Universite de la Mediterrannee, France Pedro Real, University of Sevilla, Spain Sincerely, The Organizing Committee of ATMCS III From rrosebru@mta.ca Mon Mar 17 10:43:32 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Mar 2008 10:43:32 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JbFZq-0006Df-Ak for categories-list@mta.ca; Mon, 17 Mar 2008 10:40:34 -0300 To: LICS List From: Kreutzer + Schweikardt Subject: categories: LICS Newsletter 114 Date: Mon, 17 Mar 2008 11:43:17 +0100 (CET) Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 88 Newsletter 114 March 14, 2008 ******************************************************************* * Past issues of the newsletter are available at http://www.informatik.hu-berlin.de/lics/newsletters/ * Instructions for submitting an announcement to the newsletter can be found at http://www.informatik.hu-berlin.de/lics/newsletters/inst.html * To unsubscribe, send an email with "unsubscribe" in the subject line to lics@informatik.hu-berlin.de ******************************************************************* TABLE OF CONTENTS * ANNOUNCEMENTS LICS 2008 - List of accepted papers * CONFERENCES AND WORKSHOPS SECRET 2008 - Call for Papers AiML-2008 - Third Call for papers SPIN 2008 - Final Call for Papers IMLA 2008 - Call for Papers COMPULOG/ALP Summer School VERIFY 2008 - Call for Papers ICLP 2008 - Call for Papers LASH 2008 - Call for Papers * POSITIONS POSTDOC AND PROGRAMMER POSITIONS - University of New South Wales, Austral= ia LOGIC IN COMPUTER SCIENCE (LICS) 2008 List of Accepted Papers - Klaus Aehlig and Arnold Beckmann. On the Computational Complexity of Cut-Reduction - Christian Urban, James Cheney and Stefan Berghofer. Mechanising the Metatheory of LF - Matthew Hague, Andrzej Murawski, Luke Ong and Olivier Serre. Collapsible Pushdown Automata and Recursion Schemes - Francois Pottier. Hiding Local State in Direct Style: A Higher-Order Anti-Frame Rule - Christel Baier, Nathalie Bertrand, Patricia Bouyer, Thomas Brihaye and = Marcus Groesser. Almost-Sure Model Checking of Infinite Paths in One-Clock Timed Automat= a - Taolue Chen and Wan Fokkink. On the Axiomatizability of Impossible Futures: Preorder versus Equivale= nce - Ivan Lanese, Jorge A. Perez, Davide Sangiorgi and Alan Schmitt. On the Expressiveness and Decidability of Higher-Order Process Calculi - Mikolaj Bojanczyk, Luc Segoufin and Howard Straubing. Piecewise Testable Tree Languages - Barnaby Martin, Florent Madelaine and Hubie Chen. Quantified Constraints and Containment Problems - Marc de Falco. The Geometry of Interaction of Differential Interaction Nets - Soren B. Lassen and Paul Blain Levy. Typed Normal Form Bisimulation for Parametric Polymorphism - Arnaud Carayol, Matthew Hague, Antoine Meyer, Luke Ong and Olivier Serr= e. Winning Regions of Higher-Order Pushdown Games - James F. Lynch. A Logical Characterization of Individual-Based Models - Greg Hjorth, Bakhadyr Khoussainov, Antonio Montalb\'an and Andre Nies. From Automatic Structures to Borel Structures - David Duris. Hypergraph Acyclicity and Extension Preservation Theorems - Pierre Chambart and Philippe Schnoebelen. The Ordinal Recursive Complexity of Lossy Channel Systems - Soren Riis. On the Asymptotic Nullstellensatz and Polynomial Calculus Proof Complex= ity - Vineet Kahlon. Parameterization as Abstraction: A Tractable Approach to the Dataflow A= nalysis of Concurrent Programs - Roberto Maieli and Olivier Laurent. Local Cut Elimination for Monomial MALL Proof Nets - Adri=C3=A0 Gasc=C3=B3n, Guillem Godoy and Manfred Schmidt-Schauss. Context Matching for Compressed Terms - Andrzej Murawski. Reachability Games and Game Semantics: On Comparing Nondeterministic Pr= ograms - Emmanuel Beffara. An Algebraic Process Calculus - Makoto Tatsuta. Types for Hereditary Permutators - Martin Grohe. Definable Tree Decompositions - Catarina Carvalho, Victor Dalmau and Andrei Krokhin. Caterpillar Duality for Constraint Satisfaction Problems - Olivier Delande and Dale Miller. A Neutral Approach to Proof and Refutation in MALL - Virgile Mogbil and Paulin Jacob=C3=A9 de Naurois. Correctness of Multiplicative Additive Proof Structures is NL-Complete - Guillaume Burel. A First-Order Representation of Pure Type Systems using Superdeduction - Tomas Brazdil, Jan Kretinsky, Antonin Kucera and Vojtech Forejt. The Satisfiability Problem for Probabilistic CTL - Abbas Edalat. Weak Topology and Differentiable Operator for Lipschitz Maps - Gordon Plotkin and Matija Pretnar. A Logic for Algebraic Effects - Agata Ciabattoni, Nikolaos Galatos and Kazushige Terui. From Axioms to Analytic Rules in Nonclassical Logics - Rohit Chadha, A. Prasad Sistla and Mahesh Viswanathan. On the Expressiveness and Complexity of Randomization in Finite State M= onitors - Victor Dalmau and Benoit Larose. Maltsev + Datalog -> Symmetric Datalog - Sam Staton. General Structural Operational Semantics through Categorical Logic - Carsten Sch=C3=BCrmann and Jeffrey Sarnat. Structural Logical Relations - Andrew Gacek, Dale Miller and Gopalan Nadathur. Combining Generic Judgments with Recursive Definitions - Marcelo Fiore. Second-Order and Dependently-Sorted Abstract Syntax - Daniel R Licata, Noam Zeilberger and Robert Harper. Focusing on Binding and Computation 3RD INTERNATIONAL WORKSHOP ON SECURITY AND REWRITING TECHNIQUES (SecReT'08) http://www.dsic.upv.es/workshops/secret08 Sunday, June 22, 2008, Pittsburgh, USA Affiliated workshop of the 21st IEEE Computer Security Foundations Symposium (CSF) and the 23rd IEEE Symposium on Logic In Computer Science (LICS) * IMPORTANT DATES Abstract Submission March 31, 2008 Full Paper Submission April 6, 2008 Acceptance Notification May 12, 2008 Camera Ready May 26, 2008 Workshop June 22, 2008 * SCOPE The aim of this workshop is to bring together rewriting researchers and security experts, in order to foster their interaction and develop future collaborations in this area, provide a forum for presenting new ideas and work in progress, and enable newcomers to learn about current activities in this area. The workshop focuses on the use of rewriting techniques in all aspects of security. Specific topics include: authentication, encryption, access control and authorization, protocol verification, specification of policies, intrusion detection, integrity of information, control of information leakage, control of distributed and mobile code, etc. * Previous instances of SecRet were held in 2006 (S. Servolo, Venice, Italy), and 2007 (Paris, France). * LOCATION SecReT'08 will be held at Carnegie Mellon University in Pittsburgh, Pennsylvania, USA. The workshop is associated with the 21st IEEE Computer Security Foundations Symposium (CSF'08) and the 23rd IEEE Symposium on Logic in Computer Science (LICS'08). * SUBMISSION PROCEDURE Submission is web-based via a link available in the main web page. Submissions must be received by April 6, 2008. In addition, a title and abstract must be submitted by March 31, 2008. Submitted papers should be at most 15 pages in the ENTCS style, and should include an abstract and the author's information. See the author's instructions of ENTCS style at http://www.entcs.org. * PUBLICATION Accepted papers will be published in a preliminary volume available during the workshop. After the workshop, a final version of the proceedings will be published in the Elsevier series Electronic Notes in Theoretical Computer Science (ENTCS). * INVITED SPEAKERS Hubert Comon Cachan, France Jonathan Millen MITRE, USA * PROGRAM CO-CHAIRS Daniel Dougherty Worcester Polytechnic Institute, USA Santiago Escobar Technical University of Valencia, Spain * PROGRAM COMMITTEE Pierpaolo Degano Pisa, Italy Daniel Dougherty Worcester, USA Santiago Escobar Valencia, Spain Maribel Fernandez King's College London, UK Thomas Genet IRISA Rennes, France Joshua Guttman MITRE, USA Catherine Meadows NRL, USA Monica Nesi L'Aquila, Italy Michael Rusinowitch Lorraine, France Ralf Treinen Paris-7, France ADVANCES in MODAL LOGIC (AiML-2008) 9-12 September 2008, LORIA, Nancy, France http://aiml08.loria.fr * DEADLINE: 31 March 2008 - SITE OPEN FOR SUBMISSIONS Advances in Modal Logic is an initiative aimed at presenting an up-to-date picture of the state of the art in modal logic and its many applications. The initiative consists of a conference series together with volumes based on the conferences. * AiML-2008 is the seventh conference in the series. * TOPICS We invite submission on all aspects of modal logics, including the following: - history of modal logic - philosophy of modal logic - applications of modal logic - computational aspects of modal logic + complexity and decidability of modal and temporal logics + modal and temporal logic programming + model checking + theorem proving for modal logics - theoretical aspects of modal logic + algebraic and categorical perspectives on modal logic + coalgebraic modal logic + completeness and canonicity + correspondence and duality theory + many-dimensional modal logics + modal fixed point logics + model theory of modal logic + proof theory of modal logic - specific instances and variations of modal logic + description logics + dynamic logics and other process logics + epistemic and deontic logics + modal logics for agent-based systems + modal logic and game theory + modal logic and grammar formalisms + provability and interpretability logics + spatial and temporal logics + hybrid logic + intuitionistic logic + monotonic modal logic + substructural logic Papers on related subjects will also be considered. * INVITED SPEAKERS Invited speakers at AiML-2008 will include the following: - Mai Gehrke, Radboud Universiteit Nijmegen http://www.math.ru.nl/~mgehrke/ - Guido Governatori, The University of Queensland http://www.itee.uq.edu.au/~guido/ - Agi Kurucz, King's College London http://www.dcs.kcl.ac.uk/staff/kuag/ - Lawrence Moss, Indiana University http://www.indiana.edu/~iulg/moss/ - Michael Zakharyaschev, Birkbeck College http://www.dcs.bbk.ac.uk/~michael/ * PAPER SUBMISSIONS In a change from previous AiML's, there will be two types of paper: (1) Full papers for publication and presentation at the conference. (2) Abstracts for short presentation only. Both types of paper should be submitted electronically using the submission page at http://www.easychair.org/AiML08/ The online submission system is now open. The submission deadline is 31 March 2008. * PROGRAMME COMMITTEE Alessandro Artale (Free University of Bolzano, Italy) Philippe Balbiani (IRIT, Toulouse, France) Alexandru Baltag (University of Oxford, UK) Guram Bezhanishvili (New Mexico State University, USA) Patrick Blackburn (LORIA, France) Stephane Demri (CNRS, Cachan, France) Melvin Fitting (City University of New York, USA) Guido Governatori (University of Queensland, Australia) Silvio Ghilardi (University of Milano, Italy) Valentin Goranko (University of the Witwatersrand, South Africa) Rajeev Gore (The Australian National University, Australia) Andreas Herzig (IRIT, Toulouse, France) Ian Hodkinson (Imperial College London, UK) Ramon Jansana (University of Barcelona, Spain) Alexander Kurz (University of Leicester, UK) Carsten Lutz (Dresden University of Technology, Germany) Edwin Mares (Victoria University of Wellington) Larry Moss (Indiana University, USA) Dirk Pattinson (Imperial College London, UK) Mark Reynolds (University of Western Australia, Australia) Ildiko Sain (Hungarian Academy of Sciences) Ulrike Sattler (University of Manchester, UK) Renate Schmidt (University of Manchester, UK) Jerry Seligman (University of Auckland, New Zealand) Valentin Shehtman (Moscow State University, Russia) Nobu-Yuki Suzuki (Shizuoka University, Japan) Yde Venema (ILLC, University of Amsterdam, The Netherlands) Heinrich Wansing (Dresden University of Technology, Germany) Frank Wolter (University of Liverpool, UK) Michael Zakharyaschev (Birkbeck College, London, UK) * IMPORTANT DATES Submission deadline: 31 March 2008 Acceptance notification: 31 May 2008 Final version of full papers due: 30 June 2008 Conference: 9-12 September 2008 * CONFERENCE LOCATION Advances in Modal Logic 2008 will be held at LORIA (Laboratoire Lorrain de Recherche en Informatique et ses Applications) in Nancy, in the Lorraine, in the east of France. * FURTHER INFORMATION Information about AiML-2008 will be available at the conference website: http://aiml08.loria.fr 15TH INT. SPIN WORKSHOP ON MODEL CHECKING OF SOFTWARE (SPIN) Final Call for Papers August 10-12, 2008 University of California Los Angeles, USA http://compilers.cs.ucla.edu/spin08 * Aim and Scope: The SPIN workshop is a forum for practitioners and researchers interested in state space-based techniques for the validation and analysis of software systems. Theoretical techniques and empirical evaluations based on explicit representations of state spaces, as implemented in the SPIN model checker or other tools, or techniques based on combination of explicit representations with other representations, are the focus of this workshop. We particularly welcome papers describing the development and application of state space exploration techniques in testing and verifying security-critical software, enterprise and web applications, embedded software, and other interesting software platforms. The workshop aims to encourage interactions and exchanges of ideas with all related areas in software engineering. * Invited speakers: - Matthew Dwyer (University of Nebraska) - Daniel Jackson (MIT) - Shaz Qadeer (Microsoft Research) - Wolfram Schulte (Microsoft Research) - Yannis Smaragdakis (University of Oregon) * Important Dates and Deadlines: Deadline for submission of full papers: April 2, 2008 Notification of acceptance/rejection: May 10, 2008. Deadline for final version of accepted papers: May 28, 2008. Workshop: August 10-12, 2008. * Important Dates and Deadlines: Deadline for submission of full papers: April 2, 2008 Notification of acceptance/rejection: May 10, 2008. Deadline for final version of accepted papers: May 28, 2008. Workshop: August 10-12, 2008. * Topics of Interest: - Algorithms and storage methods for explicit state model checking - Directed model checking using heuristics - Parallel or distributed model checking using multi-core or multiple computers - Techniques for dealing with infinite state spaces - Model checking of timed and probabilistic systems - Abstraction and the use of static analysis to reduce state spaces - Combinations of enumerative and symbolic techniques - Analysis for modeling languages, including SE languages (UML,...) - New property specification languages, including new forms of temporal logic - Model checking of programming languages and code analysis - Automated testing using model checking techniques - Derivation of invariants, test cases, or other useful information from state spaces - Combination of model-checking techniques with other analysis techniques - Modularity and compositionality - Comparative studies, including to other model checking techniques - Case studies of interesting systems or with interesting results - Theoretical and algorithmic foundations of model-checking based analysi= s - Engineering and implementation of model-checking tools and platforms - Insightful surveys or historical accounts on topics of relevance to SPIN workshops * Organization: General Chair: Jens Palsberg (UC Los Angeles, USA) Programme Chairs: - Klaus Havelund (NASA JPL/Caltech., USA) - Rupak Majumdar (UC Los Angeles, USA) * Programme Committee: Christel Baier (Bonn, Germany) Dragan Bosnacki (Eindhoven, Netherlands) Lubos Brim (Brno, Czech) Stefan Edelkamp (Dortmund, Germany) Dawson Engler (Stanford, USA) Kousha Etessami (Edinburgh, UK) Susanne Graf (Verimag, France) John Hatcliff (Kansas State Univ., USA) Gerard Holzmann (NASA JPL, USA) Franjo Ivancic (NEC, USA) Sarfraz Khurshid (UT Austin, USA) Kim Larsen (Aalborg, Denmark) Madan Musuvathi (Microsoft, USA) Joel Ouaknine (Oxford, UK) Corina Pasareanu (NASA Ames, USA) Doron Peled (Warwick, UK) Paul Pettersson (Malardalen, Sweden) Koushik Sen (Berkeley, USA) Natasha Sharygina (Lugano, Switzerland) Eran Yahav (IBM, USA) FOURTH INTERNATION WORKSHOP ON INTUITIONISTIC MODAL LOGIC AND APPLICATIONS = (IMLA'08) http://www.cs.bham.ac.uk/~vdp/IMLA08.html A LICS'08 affiliated workshop Pittsburgh, Pennsylvania, June 23, 2008 * Constructive modal logics and type theories are of increasing foundationa= l and practical relevance in computer science. Applications are in type disciplines for programming languages, and meta-logics for reasoning abou= t a variety of computational phenomena. Theoretical and methodological issues center around the question of how the proof-theoretic strengths of constructive logics can best be combined with the model-theoretic strengths of modal logics. Practical issues center around the question which modal connectives with associated laws or proof rules capture computational phenomena accurately and at the right level of abstraction. This workshop will bring together designers, implementers, and users to discuss all aspects of intuitionistic modal logics and type theories. * Topics include, but are not limited to: - applications of intuitionistic necessity and possibility - monads and strong monads - constructive belief logics and type theories - applications of constructive modal logic and modal type theory to formal verification, foundations of security, abstract interpretation, and program analysis and optimization - modal types for integration of inductive and co-inductive types, higher= \ - order abstract syntax, strong functional programming - models of constructive modal logics such as algebraic, categorical, Kripke, topological, and realizability interpretations - notions of proof for constructive modal logics - extraction of constraints or programs from modal proofs - proof search methods for constructive modal logics and their implementations * The workshop continues a series of previous LICS-affiliated workshops, which were held as part of FLoC'99, Trento, Italy and of FLoC'02, Copenhagen, Denmark. * IMPORTANT DATES: Submission: April 25, 2008 Notification: May 23, 2008 Final papers due: June 7, 2008 Workshop Date: June 23, 2008 * PROGRAM COMMITTEE: Gavin Bierman (Microsoft, UK) Valeria de Paiva (PARC, USA) Michael Mendler (Bamberg, DE) Aleks Nanevski (Microsoft, UK) Brigitte Pientka (McGill, CA) Eike Ritter (Birmingham, UK) * INVITED SPEAKERS: Frank Pfenning (CMU, USA) Torben Brauner (Roskilde, RK) * CONTACTS Valeria de Paiva Aleks Nanevski PARC, Palo Alto Research Center Microsoft Research paiva@parc.xeroc.com aleksn@microsoft.com 3RD INTERNATIONAL COMPULOG/ALP SUMMER SCHOOL ON LOGIC PROGRAMMING AND COMPU= TATIONAL LOGIC Sponsored by CRA-W, CDC, ALP, Compulog Americas, NMSU http://www.cs.nmsu.edu/~ipivkina/compulog.htm New Mexico State University Las Cruces, NM, USA July 24-27, 2008 * The third international summer school in Logic Programming and Computation Logic will be held on the campus of New Mexico State University in beautiful Las Cruces, New Mexico. The summer school is intended for graduate students, post-doctoral students, young researchers, and programmers interested in constraints, logic programming, computational logic and their applications. The lectures will be given by internationally renowned researchers who have made significant contributions to the advancement of these disciplines. The summer school is a good opportunity for quickly acquiring background knowledge on important areas of computational logic. The summer school is especially directed to Ph.D. students who are just about to start research. Exceptional undergraduate students in their senior year are also encouraged to attend. * The summer school will consist of six 1/2 day tutorials on the following topics: - Theoretical Foundations of Logic Programming [Miroslaw Truszczynski, U. of Kentucky] - Answer Set Programming [Torsten Schaub, U. of Potsdam] - Implementation and Execution Models for Logic Programming [Manuel Hermenegildo, Polytechnic Univ. of Madrid] - Logic Programming and Multi-agent Systems [Francesca Toni, Imperial College] - Foundations of Constraint and Constraint Logic Programming [TBA] - Foundations of Semantic Web and Computational Logic [Sheila McIlraith, University of Toronto] * Registration Due to the limit on the number of slots available, we invite interested student to submit an application for admission to the summer school composed of the following items: 1. a one page statement of interest, explaining your research background and what you expect to gain from the summer school 2. a short (2-page) vitae * Applications should be submitted in electronic form to: epontell@cs.nmsu.edu and ipivkina@cs.nmsu.edu * All submissions will be acknowledged with an email. If you do not receive acknowledgement within 3 working days, please email Enrico Pontelli (epontell@cs.nmsu.edu). * Important dates Requests for student grants: April 15, 2008; Application for Admission: April 25, 2008; Notification of Admission and grants: May 1st, 2008; Summer School: July 24-27, 2008 * Organizers Enrico Pontelli, New Mexico State University, USA Inna Pivkina, New Mexico State University, USA Karen Villaverde, New Mexico State University, USA Son Cao Tran, New Mexico State University, USA VERIFY'08 - 5th International Verification Workshop Call for Papers August 10-11, 2008, Sydney, Australia http://www.uni-koblenz.de/~beckert/verify08/ * The VERIFY workshop series aims at bringing together people who are inter= ested in the development of safety and security critical systems, in formal met= hods, in the development of automated theorem proving techniques, and in the development of tool support. Practical experiences gained in realistic verifications are of interest to the automated theorem proving community = and new theorem proving techniques should be transferred into practice. The overall objective of the VERIFY workshops is to identify open problems an= d to discuss possible solutions under the theme What are the verification problems? What are the deduction techniques? * The scope of VERIFY includes topics such as: + ATP techniques in verification + Information flow control + Case studies (specif. & verific.) + Refinement & Decomposition + Combination of verification tools + Reliability of mobile compu= ting + Integration of ATPs and CASE-tools + Reuse of specifications & p= roofs + Compositional & modular reasoning + Management of change + Experience reports on using verification + Safety-critical systems + Gaps between problems and techniques + Security models + Formal methods for fault tolerance + Tool support for formal met= hods * Important dates: Extended Abstract Submission Deadline: May 15, 2008 Extended Paper Submission Deadline: May 22, 2008 Notification of acceptance: June 25, 2008 Final version due: July 10, 2008 Workshop: August 10-11, 2008 24TH INTERNATIONAL CONFERENCE ON LOGIC PROGRAMMING (ICLP'08) First call for papers Udine, Italy, December 9th-13th, 2008 http://iclp08.dimi.uniud.it * CONFERENCE SCOPE Since the first conference held in Marseilles in 1982, ICLP has be= en the premier international conference for presenting research in logic progra= mming. Contributions (papers, position papers, and posters) are sought in all ar= eas of logic programming including but not restricted to: - Theory: Semantic Foundations, Formalisms, Nonmonotonic Reasoning, Knowledge Representation. - Implementation: Compilation, Memory Management, Virtual Machines, Paral= lelism. - Environments: Program Analysis, Program Transformation, Validation= and Verification, Debugging, Profiling, Integration. - Language Issues: Extensions, Integration with Other Paradigms, Concur= rency, Modularity, Objects, Coordination, Mobility, Higher Order, Types, M= odes, Programming Techniques. - Related Paradigms: Abductive Logic Programming, Inductive Logic Progra= mming, Constraint Logic Programming, Answer-Set Programming. - Applications: Databases, Data Integration and Federation, So= ftware Engineering, Natural Language Processing, Web and Semantic Web, Ag= ents, Artificial Intelligence, Bioinformatics * The three broad categories for submissions are: (1) Technical papers, providing novel research contributions, innova= tive perspectives on the field, and/or novel integrations across diffe= rent areas; (2) Application papers, describing innovative uses of logic program= ming technology in real-world application domains; (3) Posters, ideal for presenting and discussing current work, not yet r= eady for publication, for PhD thesis summaries and research project overv= iews. * WORKSHOPS The ICLP'08 program will include several workshops. They are perhaps th= e best place for the presentation of preliminary work, novel ideas, and ne= w open problems to a more focused and specialized audience. Workshops also pro= vide a venue for presenting specialised topics and opportunities for int= ensive discussions and project collaboration in any areas related to = logic programming, including cross-disciplinary areas. * IMPORTANT DATES Papers Posters Abstract submission deadline June 2nd n/a Submission deadline June 9th August 15th Notification of authors August 1st September 1st Camera-ready copy due September 15th September 15th 20 Years of Stable Models TBA Doctoral Consortium TBA Workshop Proposals June 2nd Early-bird Registration TBA Conference December 9-13, 2008 * Program Committee: Salvador Abreu Sergio Antoy Pedro Barahona Chitta Baral Gerhard Brewka Manuel Carro Michael Codish Alessandro Dal Palu' Bart Demoen Agostino Dovier John Gallagher Michael Gelfond Carmen Gervet Gopal Gupta Manuel Hermenegildo Andy King Michael Maher Juan Moreno Navarro Alberto Pettorossi Brigitte Pientka Gianfranco Rossi Fariba Sadri Vitor Santos Costa Tran Cao Son Paolo Torroni Frank Valencia Mark Wallace WORKSHOP ON LOGIC AND SEARCH (LaSh08) Computation of structures from declarative descriptions Call For Papers Leuven, Belgium, November 6-7, 2008 http://www.cs.kuleuven.be/~dtai/LaSh08 * IMPORTANT DATES Submission: August 15, 2008 Notification: September 15, 2008 Workshop: November 6-7, 2008 * SCOPE In many real-life problems, we search for objects of complex nature -- plans, schedules, assignments. Such objects are often represented as (finite) structures, which are implicitly specified by means of theories in some logic. Thus, languages are needed to describe structures, and algorithms to extract them from these implicit descriptions. Propositional Satisfiability (SAT), Constraint Programming (CP), and Answer Set Programming (ASP) are arguably the three most prominent areas that develop such languages and techniques. Each of these areas has been proposed as a declarative programming approach to solving NP-complete combinatorial problems. Such problems abound in computer science, engineering, operations research computational biology and other fields. In many cases, progress is limited by the difficulty of designing implicit representations of structures (modeling), which hinders common acceptance of the aproach, and the inability to solve sufficiently large instances of the problems in practical time bounds (search algorithms). Therefore, these three areas have as a major goal the development of practical modeling languages and methodologies that support the modeling, and algorithms and tools for efficient problem solving. Despite the similar goals of these areas, in many respects SAT, ASP and CP develop as three independent disciplines, focusing on rather different particular problems or questions. There are few, if any, researchers who are experts in all three areas. To date, we are not aware of any meeting which specifically aims at bringing these three areas together. * Objectives LaSh08 aims to offer a discussion forum for research in SAT, ASP and CP that focuses on the computation of structures from declarative descriptions. We invite contributions on modeling languages, methodologies, theoretical analysis, techniques, algorithms and systems. The forum is an occasion to exchange ideas on the state-of-the-art; to discuss specific technical problems; to formulate challenges and opportunities ahead; to analyse differences and simularities between the different areas; to study opportunities for synergy and integration. * In particular, we would like to foster exchange at least on the following topics: - integrations of SAT, ASP and/or CP technologies - comparisons of modeling languages - criteria for choice of modeling languages (for modeling convenience or efficiency) - new algorithm directions - efficient modeling strategies - new applications - complexity results, tractable subsets - completeness results (e.g. capturing complexity classes) - methods for taking advantage of tractability results - solver implementation techniques, - algorithms for grounding - modeling languages and constructs (aggregates, global constraints,..) - search control and heuristics in the context of model generation - symmetry breaking in model construction - optimisation problems in model construction: - languages for optimality criteria; - algorithms for computing optimal models * Systems and Tools: LaSh08 will also provide an opportunity for presentation of implemented systems and tools at a demo session. Thus, we invite submissions of systems and tools that reflect the above ideas, and aim at facilitating declarative problem solving, and making it practical and used. * Program Committee - Peter Baumgartner, The Australian National University - Francesco Calimeri, University of Calabria - Thomas Eiter, Vienna University of Technology - Wolfgang Faber, University of Calabria - Pierre Flener, Uppsala University - Alan Frisch, University of York - Enrico Giunchiglia, University of Genova - Daniel LeBerre, Universite d'Artois - Fangzen Lin, Hong kong University of Science and Technology - Ines Lynce, Universidade Tecnica de Lisboa - Tony Mancini, Sapienza Universita di Roma - Victor Marek, University of Kentucky - David Mitchell, Simon Fraser University - Pierre Marquis, Universite d'Artois - Ilkka Niemela, Helsinki University of Technology - Karem Sakallah, University of Michigan - Torsten Schaub, University of Potsdam - Barry O'Sullivan, University College Cork - Eugenia Ternovska Simon Fraser University - Mirek Truszcznski, University of Kentucky - Pascal Van Hentenryck, Brown University - Toby Walsh, University of New South Wales * Local organisation - Marc Denecker, K.U.Leuven - Joost Vennekens, K.U.Leuven * Location The conference will take place in the Beguinage of Leuven, Belgium. Leuven is an old flemish town, hosting the oldest university of the lower countries. The Beguinage is a medieval city in the city, where the beguines lived together to form a religious community. The Beguinage is recognized as a Unesco World Heritage site. POSTDOC AND PROGRAMMER POSITIONS ARE AVAILABLE IN THE SCHOOL OF COMPUTER SC= IENCE AND ENGINEERING, UNIVERSITY OF NEW SOUTH WALES, AUSTRALIA. * The positions are associated to an Australian Research Council Linkage Grant funded project "Model Checking Logics of Knowledge and Probability in Pursuit-Evasion Games". The research will involve the development of model checking techniques for the logic of knowledge, probability and time, and their evaluation in the partner's application: pursuit-evasion games motivated from search and rescue mission planning problems. * For details, see http://www.cse.unsw.edu.au/~meyden/positions/. From rrosebru@mta.ca Mon Mar 17 10:43:32 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Mar 2008 10:43:32 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JbFVj-0005LX-S0 for categories-list@mta.ca; Mon, 17 Mar 2008 10:36:19 -0300 Date: Mon, 17 Mar 2008 01:22:48 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: Categories list Subject: categories: Re: mathematical articles in online encyclopedias Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 89 Paul Taylor wrote: > It was Vaughan Pratt who first introduced the Wikipedia thread, in > response to someone who said that he hadn't heard of Heyting algebras. > ... > I have changed the Subject line because Wikipedia is not the only > site of its kind. Anyone thinking of writing for it should perhaps > also consider: > - citizendium.org - which looks like Wikpedia because it is run by > the latter's co-founder and now unperson; citizendium has a strict > policy of using real names and qualifications; > - planetmath.org - in which authors "own" the pages that they have > written until they've demonstrably abandoned them; > - mathworld.wolfram.com - beware that this is owned by Wolfram. > This is off-topic only to the extent that it concerns a publication medium that is as open to articles on the animal liberation movement as it is to those on toposes, subobject classifiers (separate from toposes!) and abelian categories. Wikipedia's flexibility has its pros and cons. While it is potentially as corruptible as communism, by its nature it is dominated by the intelligentsia rather than either the bourgeoisie or the proletariat. Common sense being uniformly albeit sparsely distributed among all three classes, there is no apriori reason why domination of this kind should handicap it any more than its competitors. A significant advantage of Wikipedia is that it was there first (among those open encyclopedias that have amounted to anything) and has become the Microsoft of its genre much faster than Microsoft itself. The fact that some academics remain skeptical of its quality is not in practice a serious differentiator from its competitors. Articles vary widely in quality. I'm presently involved in a dispute over replacing an account of Boolean algebra at http://en.wikipedia.org/wiki/Boolean_logic with my version of that story at http://en.wikipedia.org/wiki/Boolean_algebra_%28introduction%29 . The latter did not exactly spring full grown from my brow---I started out with http://en.wikipedia.org/wiki/Boolean_algebras_canonically_defined as a kind of protest against what I perceived as outdated and parochial views of the subject but then realized that this was pretty avant garde compared to what was needed and toned it down to http://en.wikipedia.org/wiki/Boolean_algebra_%28logic%29 . But pretty soon it became clear to me that this too was pitched at too high a level for Wikipedia and I tried again with http://en.wikipedia.org/wiki/Boolean_algebra_%28introduction%29 . I'm sure that can be simplified too, but the author of http://en.wikipedia.org/wiki/Boolean_logic has utterly failed to convince me that his account is the way to go. Meanwhile I've wrestled with other appalling accounts of topics such as residuated lattices (I completely replaced an article that in effect defined them to be Heyting algebras) and relation algebras (replacing an article that faithfully transcribed all the metamathematical Greek letters in Tarski and Givant's "Set Theory without Variables" in favor of notation more appropriate to an account of a variety). Then there's articles on dynamic logic, Zhegalkin polynomials, and Zhegalkin himself. Another timesink is the pseudoscience that well-intentioned but under-calibrated editors have to struggle with, such as the Wolfram prize for a supposedly tiny universal Turing machine, and Burgin's notion of "super-recursive algorithm" as his proposed counterexample to the Church-Turing thesis. In short, much like the real world, which still hasn't converged on Utopia despite trying hard and wishing harder. Wikipedia and the world are difficult but vibrant and growing communities and I hold out great hopes for the future of both. Vaughan From rrosebru@mta.ca Mon Mar 17 10:43:32 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Mar 2008 10:43:32 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JbFYy-00064d-Gt for categories-list@mta.ca; Mon, 17 Mar 2008 10:39:40 -0300 Date: Mon, 17 Mar 2008 11:22:35 +0100 From: Joachim Kock Subject: categories: HOCAT 2008 To: categories@mta.ca MIME-version: 1.0 Content-type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 90 This is a reminder of the conference HOCAT 2008 Homotopy Structures in Geometry and Algebra; Derived Categories, Higher Categories which will be held from June 30 to July 5, 2008, at the Centre de Recerca Matem=E0tica, Barcelona. This is an event within the CRM thematic year on Homotopy=20 Theory and Higher Categories (http://www.crm.cat/hocat/). The following have agreed to speak at the conference: John Baez (University of California at Riverside) Paul Balmer (University of California at Los Angeles) David Benson (University of Aberdeen) Julia Bergner (Kansas State University) Tom Bridgeland (University of Sheffield) S=F8ren Galatius (Stanford University) Ezra Getzler (Northwestern University) Mikhail Kapranov (Yale University), to be confirmed Ralf Meyer (Georg-August Universit=E4t G=F6ttingen), to be confirmed Charles Rezk (University of Illinois at Urbana) Bertrand To=EBn (Universit=E9 Paul Sabatier, Toulouse) Michel Van den Bergh (Hasselt University) A limited number of slots are available for contributed talks. Prospective speakers should submit an abstract to any of the organisers before April 30. For registration (deadline May 31) and further information about the conference, see http://www.crm.cat/HOCAT2008/. We look forward to seeing you in Barcelona. The organisers, Carles Casacuberta Andr=E9 Joyal Joachim Kock Amnon Neeman Frank Neumann From rrosebru@mta.ca Mon Mar 17 10:43:32 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Mar 2008 10:43:32 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JbFSk-0004pC-22 for categories-list@mta.ca; Mon, 17 Mar 2008 10:33:14 -0300 Date: Sun, 16 Mar 2008 23:36:28 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories list Subject: categories: The internal logic of a topos Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 91 As I understand the internal logic of a topos it consists of certain morphisms from finite powers of Omega to Omega. In the case of Set it consists of all such morphisms. For which toposes is this not the case, and for those how are the morphisms that do belong to the internal logic best characterized? I do hope it's not necessary to start from the notion of an internal Heyting algebra, that sounds so counter to mathematical practice and intuition. If the internal logic consists of precisely those morphisms preserved by geometric morphisms this will give me the necessary motivation to go to the mats with geometry. Vaughan From rrosebru@mta.ca Mon Mar 17 10:50:30 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Mar 2008 10:50:30 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JbFiZ-00005x-8A for categories-list@mta.ca; Mon, 17 Mar 2008 10:49:35 -0300 Date: Mon, 17 Mar 2008 14:36:56 +0300 From: Jawad Abuhlail Subject: categories: Re: The Category of Semimodules over Semirings To: categories@mta.ca MIME-version: 1.0 Content-type: text/plain; charset=us-ascii Content-transfer-encoding: 7BIT Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 92 Dear Prof. Linton, Many thanks for your comments about the existence of coequalizers in categories of semimodules. What I mentioned (that the category of left semimodules over an arbitrary semiring has in general no coequalizers) was due to CONFUSION caused by the way some results in "M. Takahashi, Completeness and $C$-co completeness of the category of semimodules. Math. Sem. Notes Kobe Univ. 10 (1982), no. 2, 551--562." are stated. In that paper, Takahashi proved that the category of left semimodules over an arbitrary semiring has $c$-coequalizers and is $c$-cocomplete (where $c : R-smod ---> C-R-smod$ denote the functor that assigns to each semimodule its associated cancellative semimodule). Indeed his proof does not exclude that this category has coequalizers as I (apparently incorrectly) stated. For your convenience, I summarize what Takahashi did in the above mentioned paper: Denote the category of left semimodules over a semirings $R$ by $R-smod$ and its full subcategory of cancellative semimodules by $C-R-smod$. Then $R-smod$ has products and equalizers, whence complete. Let $c : R-smod ---> C-R-smod$ denote the functor that assigns to each semimodule its associated cancellative semimodule. This functor is left adjoint to the embedding functor $U : C-R-smod ---> R-smod$. Then $R-smod$ has coproducts and $c$-coequalizers, whence $c$-cocomplete. The confusion is caused by his statement that "$c$-cocompleteness is a relaxation of cocompleteness" and the last Corollary in the paper, in which he deduced that the full subcategory $C-R-smod$ of CANCELLATIVE semimodules has coequalizers and is cocomplete!! Anyway, I am so grateful for your comments and would appreciate as well any comment about exactness of colimits in $C-R-smod$ and $C-R-smod$ (in the category of modules over rings, colimits are exact!!) will be highly appreciated. Wassalam, Jawad -----Original Message----- From: Fred E.J. Linton [mailto:fejlinton@usa.net] Sent: Sunday, March 16, 2008 8:43 PM To: categories@mta.ca Cc: Jawad Abuhlail Subject: Re: categories: The Category of Semimodules over Semirings On Sun, 16 Mar 2008 09:26:35 AM EDT Jawad Abuhlail wrote, in part, on the Subject: The Category of Semimodules over Semirings, > ... The category of semimodules had products, equalizers and products > (however not necessarily coequalizers). I must be missing something here. Don't the (say, left-) semimodules (over a given semiring) constitute an equationally definable class of algebras? That is, aren't they determined entirely by operations and equations? If they DO, that is, if they ARE, then the category of them all (together with their homomorphisms) must, like all such "varietal categories," have all (small) limits and colimits, and, in particular, all coequalizers. Alas, I have little else to offer. Cheers, and Happy St. Paddy's Day, -- Fred From rrosebru@mta.ca Mon Mar 17 18:15:34 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Mar 2008 18:15:34 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JbMed-0005p9-TM for categories-list@mta.ca; Mon, 17 Mar 2008 18:14:00 -0300 From: Thomas Streicher Subject: categories: trying to answer some of Paul's questions To: categories@mta.ca Date: Mon, 17 Mar 2008 16:17:03 +0100 (CET) MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 93 Dear Paul, I am trying to answer at least some of your questions. > what he might mean categorically by saying that > an external family of objects is a function I -> Ob(EE) where I is a > set unless his Ob(EE) is a "class" in either the sense of algebraic set > theory (ie an object in another category besides EE) or some similar > approach. You can't say this "categorically" as far as I can see. I am working informally on the meta-level and there both I and Ob(EE) are ome sets and it is clear what is a function from I to Set. If necessary I can also reveil the strength of my meta-level reasoning: ZFC plus the axiom that every set appears as member of some Grothendieck universe. > AST is an intellectually valid point of view on which much good > category theory has been done in the past few years. However, > Mike Shulman and I would both like to know how one might formulate > replacement WITHOUT classes. Well, for the set theorist it's impossible anyway since he needs at least the class V of all sets which, of course, is carefully hidden as the (interpretation of the) single sort over which the variables of the language of set theory range. Well, and by using first order definable properties one has a lot of further classes not given official status but which are there as interpretations of those formulas. That's precisely the starting point of AST postulating a Heyting category EE and a "universe" V in it which has enough properties for V being a model of IZF. (Well, AST in its orginal form has also the ambition to construct this class V as a quotient of a W-type.) Only when reading your most recent mail it became clear to me that your intentions are quite different from ETCS. McLarty's replacement axiom does speak about external families which are syntactically definable in the language of ETCS whereas you - as I understand it - want to stick to internal families (aka display maps). The intention of my mail from Saturday was to argue that the external notion of family is (at least) problematic in a non-wellpointed context. Using universes in toposes (as in my paper with this title) one can formulate everything without using the external families and so one can in AST. > As I said before, they claim that replacement is necessary > to construct the ordinal omega+omega, whilst this order structure can > be constructed up to isomorphism very easily without it. Of course, one can construct even larger prim.rec. orders but one cannot prove their implementation as von Neumann ordinals. And that's what they want. > Thomas Streicher uses indexed/fibred category theory to link internal > and external notions. I don't see, however, how he manages to give > the SPECIFICATION that a display map X-->N is the sequence of iterates > of the powerset on N. The formulation that I gave at the end of my > previous posting and in Remark 9.6.16 in my book does do this. But it's very easy to express this specification in the internal language of a topos namely as \forall n:N \exists i : X_{n+1} -> P(X_n) Iso(i) > the Artin gluing / Freyd cover / scone (Sierpinski cone) / logical > relations construction, which can be used to prove CONSISTENCY > of various theories, once again by comparing the term model with > the ambient universe. in PTJ's Elephant 3.6.3 (f) you find a description of sconing in a fibred setting, i.e. over an arbitrary base: if p : EE -> SS is a geom. morph. then sc_SS(EE) is just the glueing of p_* : EE -> SS that's my answer to the question of how I would define sconing relative to an arbitrary base this certainly doesn't answer you question above; I can just give an honest answer: when using glueing I always used full power on the meta level including the common identification of families of sets indexed by I and maps to I asa far as AST is concerned there is no proof yet that models of AST are stable under soning; a first attempt you can find in a paper by M. Warren Michael A. Warren. Coalgebras in a category of classes. Annals of Pure and Applied Logic, 146(1):60-71, 2007. That's what Benno van den Berg just told me when asking him about it. Sconing for the free topos with nno is of course not the issue here. But to show that models of AST are stable under sconing is subtle. The key point is that you seem to take for Set a big enough category containg a nontrivial notion of small map. Otherwise there are problems to define an object of small objects. Thomas From rrosebru@mta.ca Mon Mar 17 18:15:34 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Mar 2008 18:15:34 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JbMc6-0005XX-97 for categories-list@mta.ca; Mon, 17 Mar 2008 18:11:22 -0300 Date: Mon, 17 Mar 2008 14:40:32 +0000 (GMT) From: "Prof. Peter Johnstone" To: Categories mailing list Subject: categories: Re: The internal logic of a topos MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 94 Dear Vaughan, I don't think one can give a straight answer to this question: it all depends on what you mean by `the logic of a topos'. I presume you're thinking of the fact that, in Set, any function 2^n --> 2 is a polynomial in the Boolean operations (i.e., is the interpretation of some n-ary term in the theory of Boolean algebras). One could ask the same question about a general topos, with `Heyting' replacing `Boolean'; but the answer would mostly be `no', even for Boolean toposes. On the other hand, one might well *define* `the internal logic of a topos' as meaning the collection of all natural operations on subobjects -- that is, the collection of all morphisms \Omega^n --> \Omega. Incidentally, there is nothing unnatural or counterintuitive about `the notion of internal Heyting algebra: it is a very natural consequence of the definition of a subobject classifier, see A1.6.3 in the Elephant. Peter On Sun, 16 Mar 2008, Vaughan Pratt wrote: > As I understand the internal logic of a topos it consists of certain > morphisms from finite powers of Omega to Omega. In the case of Set it > consists of all such morphisms. For which toposes is this not the case, > and for those how are the morphisms that do belong to the internal logic > best characterized? > > I do hope it's not necessary to start from the notion of an internal > Heyting algebra, that sounds so counter to mathematical practice and > intuition. > > If the internal logic consists of precisely those morphisms preserved by > geometric morphisms this will give me the necessary motivation to go to > the mats with geometry. > > Vaughan > > > From rrosebru@mta.ca Mon Mar 17 18:16:26 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Mar 2008 18:16:26 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JbMgj-00066K-Oc for categories-list@mta.ca; Mon, 17 Mar 2008 18:16:09 -0300 From: "George Janelidze" To: "\"Categories\"" Subject: categories: Max Kelly Volume of APCS Date: Mon, 17 Mar 2008 23:11:30 +0200 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 95 Dear Colleagues, Following Max Kelly Conference in Category Theory (Cape Town, January 2008), a Special Volume of "Applied Categorical Structures" dedicated to the Memory of Max Kelly, and edited by Martin Hyland George Janelidze Michael Johnson Peter Johnstone Stephen Lack Ross Street Walter Tholen Richard Wood will be published. The content will essentially include but not limited to the work presented on the conference; the submission deadline is 1 June 2008. Please submit papers in the usual way online on the site http://www.editorialmanager.com/apcs/ There are step by step instructions on the site as to how to do this. Please only have in mind the following: 1. You have to indicate one of the guest editors to handle your paper. That is, you MUST make a choice and it MUST be out of the guest editors from the list above. 2. You MUST indicate that your paper is intended for the special issue. For that, once you have logged in and selected to submit a paper, you have also to select the "Article Type". There will be a special article type for the issue named "Special Issue Max Kelly". On behalf of the Editorial Board, George Janelidze From rrosebru@mta.ca Tue Mar 18 08:48:03 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 18 Mar 2008 08:48:03 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JbaCW-00059T-45 for categories-list@mta.ca; Tue, 18 Mar 2008 08:41:52 -0300 Date: Mon, 17 Mar 2008 20:14:58 -0500 From: "Michael Shulman" Subject: categories: Re: internal versus external To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Content-Disposition: inline References: Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 96 On Sat, Mar 15, 2008 at 11:31 AM, Thomas Streicher wrote: > It might be the case that > any "syntactically definable" external family indexed by EE(1,X) arises from > some internal family a : A -> X though I don't see how to prove it. I thought this was exactly what Colin's version of the replacement axiom says. > The reason why I doubt that models of ETCS and bZ are equivalent is that when > going from a model of ECTS to a model of bZ is that one has to restrict to the > well-founded part. At least that's what I recollect from MacLane and Moerdijk's > exposition in their book. But I am ready to believe that adding wellfoundedness > axioms to ETCS can remedy this situation. Osius uses a different construction than M&M: instead of explicitly building "membership trees" he uses objects equipped with a transitive well-founded relation to represent transitive sets, and subobjects of them to represent arbitrary sets. In general you do have to add a "transitive representation" axiom to ensure that every object of the topos can be reconstructed from the resulting model of bZ---but this is unnecessary if you also assume choice, since then every object can be well-ordered and thus admits a transitive well-founded relation. Mike From rrosebru@mta.ca Tue Mar 18 19:03:06 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 18 Mar 2008 19:03:06 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JbjkP-00079l-1R for categories-list@mta.ca; Tue, 18 Mar 2008 18:53:29 -0300 From: Thomas Streicher Subject: categories: question to Colin about uniqueness in his Replacement axiom To: categories@mta.ca Date: Tue, 18 Mar 2008 12:50:50 +0100 (CET) MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 97 Mike Shulman pointed me out a faulty formulation in my lengthy mail from last Saturday; I take the opportunity of formulating it correctly: In your Replacement axiom (p.48 of your "Philosophia" article) you psotulta the existence of a map f : S -> A such that S_x \cong x^*f for all x : 1->X. Can you prove that this f is unique up to isomorphism, i.e. that wellpointedness for maps entails wellpointedness for families? Thomas From rrosebru@mta.ca Tue Mar 18 19:03:06 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 18 Mar 2008 19:03:06 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Jbjlv-0007L8-6F for categories-list@mta.ca; Tue, 18 Mar 2008 18:55:03 -0300 Date: Tue, 18 Mar 2008 11:19:00 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: Categories mailing list Subject: categories: Re: The internal logic of a topos Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 98 Dear Peter, Your answers come as something of a relief, in that on the one hand it seemed unlikely that every morphism \Omega^n --> \Omega would arise as a Heyting polynomial, yet I didn't see why that should be taken as grounds for calling such counterexamples nonlogical. This was the nature of my concern about Heyting algebras, that they might feature somehow in the antecedent of someone's definition of logicality. I'm more than happy to have Heyting algebras arise as a natural consequence of something even more natural such the notion of topos. I've known and loved Heyting algebras much longer than I have toposes, yet I now see toposes as being prior to Heyting algebras in the causal chain of things, and would be most uncomfortable with a definition of logicality in a topos that arbitrarily took the notion of Heyting algebra as a criterion in any essential way. So I find the thought that there could be logical morphisms in a topos that aren't Heyting polynomials quite comforting, as it tends to put Heyting algebras in their place as themselves a natural *part* of the internal logic of a topos thus understood without however being the whole of it. In a private reply Andrej Bauer made the nice point, obvious in retrospect, that the logical morphisms as the morphisms \Omega^n --> \Omega shouldn't assume n is finite or even discrete, to allow quantification over any type. He also brought up the matter of higher order logic which hadn't been on my agenda but probably should be at some point. Vaughan Prof. Peter Johnstone wrote: > Dear Vaughan, > > I don't think one can give a straight answer to this question: it all > depends on what you mean by `the logic of a topos'. I presume you're > thinking of the fact that, in Set, any function 2^n --> 2 is a polynomial > in the Boolean operations (i.e., is the interpretation of some n-ary term > in the theory of Boolean algebras). One could ask the same question about > a general topos, with `Heyting' replacing `Boolean'; but the answer > would mostly be `no', even for Boolean toposes. On the other hand, one > might well *define* `the internal logic of a topos' as meaning the > collection of all natural operations on subobjects -- that is, the > collection of all morphisms \Omega^n --> \Omega. > > Incidentally, there is nothing unnatural or counterintuitive about `the > notion of internal Heyting algebra: it is a very natural consequence of > the definition of a subobject classifier, see A1.6.3 in the Elephant. > > Peter > > On Sun, 16 Mar 2008, Vaughan Pratt wrote: > >> As I understand the internal logic of a topos it consists of certain >> morphisms from finite powers of Omega to Omega. In the case of Set it >> consists of all such morphisms. For which toposes is this not the case, >> and for those how are the morphisms that do belong to the internal logic >> best characterized? >> >> I do hope it's not necessary to start from the notion of an internal >> Heyting algebra, that sounds so counter to mathematical practice and >> intuition. >> >> If the internal logic consists of precisely those morphisms preserved by >> geometric morphisms this will give me the necessary motivation to go to >> the mats with geometry. >> >> Vaughan >> >> >> > > From rrosebru@mta.ca Tue Mar 18 19:03:07 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 18 Mar 2008 19:03:07 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Jbjo3-0007W5-BJ for categories-list@mta.ca; Tue, 18 Mar 2008 18:57:15 -0300 Date: Tue, 18 Mar 2008 13:11:44 -0500 (EST) From: Michael Barr To: Categories list Subject: categories: A question on adjoints MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 99 I guess I am getting old and dumb. This question should have been a snap for me years ago. It is old fashioned, only a 1-categorical question and not about internal vs. external. Suppose F: A --> B is left adjoint to U: B --> A. Suppose a is an object of A and b, b' objects of B such that there is an equalizer a ---> Ub ===> Ub'. (The two arrows Ub to UB' are not assumed to be U of arrows from B.) Does it follow that a ---> UFa ===> UFUFa is an equalizer? The arrows are \eta a, UF\eta a and \eta UFa of course. Michael From rrosebru@mta.ca Tue Mar 18 19:03:07 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 18 Mar 2008 19:03:07 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Jbjl4-0007F4-Oh for categories-list@mta.ca; Tue, 18 Mar 2008 18:54:11 -0300 Date: Tue, 18 Mar 2008 16:18:12 +0100 From: Ugo Dal Lago MIME-Version: 1.0 To: dallago@cs.unibo.it Subject: categories: CSL 2008 - Last CfP - Deadline is March 28th Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 100 --------------------------------------------------------------------- CALL FOR PAPERS Computer Science Logic 2008 CSL 2008 17th Annual Conference of the European Association for Computer Science Logic Bertinoro (Bologna), Italy 15 - 20 September 2008 --------------------------------------------------------------------- Abstract submission: March 28, 2008 Paper submission: April 7, 2008 Author notification: May 19, 2008 --------------------------------------------------------------------- http://csl2008.cs.unibo.it --------------------------------------------------------------------- Computer Science Logic (CSL) is the annual conference of the European Association for Computer Science Logic (EACSL). The conference is intended for computer scientists whose research activities involve logic, as well as for logicians working on issues significant for computer science. Topics of interest include: automated deduction and interactive theorem proving, constructive mathematics and type theory, equational logic and term rewriting, automata and games, modal and temporal logics, model checking, logical aspects of computational complexity, finite model theory, computational proof theory, logic programming and constraints, lambda calculus and combinatory logic, categorical logic and topological semantics, domain theory, database theory, specification, extraction and transformation of programs, logical foundations of programming paradigms, verification and program analysis, linear logic, higher-order logic, nonmonotonic reasoning. Proceedings will be published in the LNCS series. Each paper accepted by the Programme Committee must be presented at the conference by one of the authors, and final copy be prepared according to Springer's guidelines. Submitted papers must be in Springer's LNCS style and of no more than 15 pages, presenting work not previously published. They must not be submitted concurrently to another conference with refereed proceedings. Any closely related work submitted by the authors to a conference or journal before March 28, 2008 must be reported to the PC chairs. Papers authored or coauthored by members of the Programme Committee are not allowed. Submitted papers must be in English and provide sufficient detail to allow the Programme Committee to assess the merits of the paper. Full proofs may appear in a technical appendix which will be read at the reviewer's discretion. The title page must contain: title and author(s), physical and e-mail addresses, identification of the corresponding author, an abstract of no more than, 200 words, and a list of keywords. ACKERMANN AWARD: The Ackermann Award is the EACSL Outstanding Dissertation Award for Logic in Computer Science. The Ackermann Award 2008 will be presented to the recipients at CSL2008. Deadline for nominations is March 15, 2008. Details at: http://www.dimi.uniud.it/~eacsl/submissionsAck.html For the three years 2007-2009, the Award is sponsored by Logitech, S.A., Romanel, Switzerland, the world's leading provider of personal peripheral= s. INVITED SPEAKERS: Luca Cardelli, Microsoft Research, Cambridge Pierre Louis Curien, PPS, Paris Jean-Pierre Jouannaud, Ecole Polytechnique, Palaiseau Wolfgang Thomas, RWTH, Aachen PROGRAM COMMITTEE: Michael Kaminski (co-chair), Technion, Haifa Simone Martini (co-chair), Universit=E0 di Bologna Zena Ariola, University of Oregon, Eugene Patrick Baillot, CNRS and Universit=E9 Paris 13 Patrick Cegielski, Universit=E9 Paris 12 Gilles Dowek, =C9cole Polytechnique, Palaiseau Amy Felty, University of Ottawa Marcelo Fiore, University of Cambridge Alan Jeffrey, Bell Labs, Alcatel-Lucent Leonid Libkin, University of Edinburgh Zoran Majkic, University of Beograd Dale Miller, INRIA-Futurs, Palaiseau Luke Ong, University of Oxford David Pym, HP Labs, Bristol and University of Bath Alexander Rabinovich, Tel Aviv University Antonino Salibra, Universit=E0 Ca' Foscari, Venezia Thomas Schwentick, Universit=E4t Dortmund Valentin Shehtman, Moscow University and King's College London Alex Simpson, University of Edinburgh Gert Smolka, Universit=E4t des Saarlandes, Saarbr=FCcken Kazushige Terui, National Institute of Informatics, Tokyo Thomas Wilke, Universit=E4t Kiel ORGANIZATION: Ugo Dal Lago, Universit=E0 di Bologna Simone Martini, Universit=E0 di Bologna --------------------------------------------------------------------- From rrosebru@mta.ca Wed Mar 19 15:18:53 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 19 Mar 2008 15:18:53 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Jc2nk-0001Ir-Pu for categories-list@mta.ca; Wed, 19 Mar 2008 15:14:12 -0300 From: Colin McLarty To: Categories list Date: Wed, 19 Mar 2008 09:02:51 -0400 MIME-Version: 1.0 Subject: categories: Re: categorical formulations of Replacement Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 101 Paul Taylor Wednesday, March 19, 2008 8:05 am Writes > I really do not have much idea of what you mean by statements like > \forall n:N \exists i : X_{n+1} -> P(X_n) Iso(i). > Nor do I understand similar statements to this in either > "An elementary theory of the category of sets (extended version)" > or "Exploring categorical structuralism" > by Bill Lawvere and Colin McLarty respectively. As to replacement in ETCS: ETCS is formulated in the first order language (with equality) of category theory. Take it as a one-sorted language (arrows) with composition C(g,f;h). It makes no principled difference for our purposes but is extremely handy to also assume constants 1 of set type (axiomatized as terminal) and "true" of function type (axiomatized as an element of a truth value set) and partially defined operators for say, pullbacks and the evaluation functions for function sets. ZF is formulated in the first order language (with equality) with set-membership epsilon. Replacement in ETCS like replacement in ZF is an axiom scheme positing one axiom for each formula of a certain form in the first order language of the theory. ZF-replacement posits one quantified axiom for each formula Rxy with two free variables (necessarily variables over sets, since that is what ZF has, and if you like you may allow other variables as parameters). The axiom for Rxy says "For any set S, if R relates each element x\in S to a unique set y then there is a set X whose elements are exactly those sets y that are R-related to some x\in S." ETCS posits one quantified axiom for each formula Rfy with f of arrow type (the axiom will say f has domain 1 so f stands for some element of a set) and y of set type. The axiom for Rfy says "For any set S, if R relates each function f:1-->S to a set y unique up to isomorphism then there is an S-indexed set of sets X-->S where the fiber over each x is isomorphic to the related set y." The apparatus of discrete fibrations applies here and no doubt to good advantage for serious work. But very little is needed in stating the axiom scheme. Let me say again that my account of replacement is just Bill's from 1965 only cast as replacement rather than reflection since people are far more familiar with replacement (and using the simplifications to ETCS that came with elementary topos theory). Colin From rrosebru@mta.ca Wed Mar 19 15:18:54 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 19 Mar 2008 15:18:54 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Jc2mT-00015Y-P2 for categories-list@mta.ca; Wed, 19 Mar 2008 15:12:53 -0300 Date: Tue, 18 Mar 2008 22:43:35 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: Categories list Subject: categories: Re: A question on adjoints Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 102 Isn't the following a counterexample? Let A = Set and let B = A\{0} (the category of nonempty sets). Let F send the empty set in A to the singleton set in B, and otherwise let F and U be the evident identity functors between A and B. Similarly let \eta and \epsilon be the identity natural transformations, except for \eta_0 which can only be the unique function from 0 to 1. Naturality of \eta and \epsilon depends on both being the identity, except for \eta_0 but that's from the initial object so all its diagrams commute. Then 0 equalizes the two arrows from U1 to U2 but \eta_0 does not equalize UF\eta a and \eta UFa since the latter two are both 1_1 in A whence they are equalized by 1. Vaughan Michael Barr wrote: > I guess I am getting old and dumb. This question should have been a snap > for me years ago. It is old fashioned, only a 1-categorical question and > not about internal vs. external. > > Suppose F: A --> B is left adjoint to U: B --> A. Suppose a is an object > of A and b, b' objects of B such that there is an equalizer > a ---> Ub ===> Ub'. (The two arrows Ub to UB' are not assumed to be U > of arrows from B.) Does it follow that a ---> UFa ===> UFUFa is an > equalizer? The arrows are \eta a, UF\eta a and \eta UFa of course. > > Michael > > From rrosebru@mta.ca Wed Mar 19 15:18:54 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 19 Mar 2008 15:18:54 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Jc2ld-0000xc-JD for categories-list@mta.ca; Wed, 19 Mar 2008 15:12:01 -0300 From: Colin McLarty To: categories@mta.ca Date: Tue, 18 Mar 2008 18:28:28 -0400 MIME-Version: 1.0 Content-Language: en Subject: categories: Re: question to Colin about uniqueness in his Replacement axiom Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 103 Thomas Streicher Tuesday, March 18, 2008 6:09 pm Wrote: > In your Replacement axiom (p.48 of your "Philosophia" article) you > psotultathe existence of a map f : S -> A such that S_x \cong x^*f > for all x : 1->X. > Can you prove that this f is unique up to isomorphism, i.e. that > wellpointedness for maps entails wellpointedness for families? Sure. It takes the axiom of choice of course, since without choice the result may be false (even two countably infinite families of countably infinite sets need not be isomorphic). It is the obvious argument by Zorn's lemma, which follows from choice: Given two families S-->A and S'-->A with corresponding fibers isomorphic, consider the set of all pairs with U a subset of A, and i an isomorphism over U from the restriction of S to the restriction of S'. By Zorn at least one of these is maximal (for the obvious ordering by inclusion) so call it . Since well-pointedness implies Boolean, U has a complement in A--which must be empty or else we could extend the isomorphism. best, Colin From rrosebru@mta.ca Wed Mar 19 15:18:54 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 19 Mar 2008 15:18:54 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Jc2n9-0001DV-Lr for categories-list@mta.ca; Wed, 19 Mar 2008 15:13:35 -0300 Mime-Version: 1.0 (Apple Message framework v624) Content-Type: text/plain; charset=US-ASCII; format=flowed Content-Transfer-Encoding: 7bit From: Paul Taylor Subject: categories: categorical formulations of Replacement Date: Wed, 19 Mar 2008 11:54:52 +0000 To: Categories list Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 104 Dear Thomas, I'm glad that we've now started to talk a common language about Replacement, and am hopeful that it will be possible to come to some agreement, but I think that we are still some way off doing so. Since you have changed the Subject: line several times, I would like first to give some help to anyone who might be trying to follow this discussion from an archive in the future, by listing the Subject: lines of recent postings. Of course, they have "categories:" and "re:" added to them. Categorial foundations categorical formulations of Replacement Heyting algebras and Wikipedia I was partly wrong internal versus external question to Colin about uniqueness in his Replacement axiom replacement and iterated powersets replacement and the gluing construction replacing set theory the axiom scheme of replacement in category theory trying to answer some of Paul's questions When I was trying to understand replacement, ten years ago and more, I found, both from my own experience and in looking that the work of others, that it is easy to fall into one of two traps: (1) lack of rigour; using the words "external" or "meta-language" may indicate this; (2) lack of force; using the word "definable" may indicate this. You can talk rigorously about external or meta- things if you first set up a two-level formal system. Examples of such systems include (1) first order logic and set theory expressed within it; (2) first order logic and category theory expressed within it; (3) a category with an internal category; (4) a fibred category containing a universe in the sense that you and I discussed in the 1980s; (5) a pretopos or arithmetic universe with a class of small maps (algebraic set theory). A large part of the explanation for ideological conflicts between mathematicians is that they work in different OUTER systems. If they can agree on the outer system, they have an arena in which to compare their INNER systems. Reading between the lines of your posting suggests that you are not completely confident yourself of the rigour of your own account. One of the uses of a two-level system is to discuss logical questions such a consistency. For example, Godel's theorem is about truth and provability, which may be seen as facts about the outer and inner systems respectively. Andre' Joyal set this up in category theory by looking at the free arithmetic universe inside an arithmetic universe. The axiom-scheme of replacement seems to be about making the inner world agree with part of the outer one. I really do not have much idea of what you mean by statements like \forall n:N \exists i : X_{n+1} -> P(X_n) Iso(i). Nor do I understand similar statements to this in either "An elementary theory of the category of sets (extended version)" or "Exploring categorical structuralism" by Bill Lawvere and Colin McLarty respectively. It would help if you were all to give more "turorial" explanations of these things, and precise internal references to relevant papers and books, because these are often lengthy and largely devoted to simpler categorical structure than replacement. We agree, I believe, that fibred methods are they way that we can express in category theory ideas that the set theorists encode as sets of sets. Thus a display map p:X-->N captures the same idea as { {{x, {x, n}} | x in X & p(x)=n } | n in N }, or whatever hieroglyphics the set theorists would use. Similarly, the idea of a functor from a small category to a large one, say F:CC-->Set, can be captured as a discrete fibration p:FF-->CC. In order to have any chance of fitting the axiom scheme of replacement into our skulls, we have to take the technology (fibred category theory, for example) as read, even though it is rather difficult and complicated itself. The problem is that most accounts add replacement as a brief footnote to a lengthy treatment of more basic topics. My book is guilty of this, and so, with all due respect, are you, I believe. With regard to the example of the iterated powerset, the statement of yours that I quoted above claims to express this, but I do not understand the language. I would like you to translate it into the usual language of category theory, ie functors, pullbacks etc. I suspect that what you will come up with is the same as in my posting about this on Sunday afternoon. I accept that I have taken the technology of fibred category theory off the shelf to do this, but ********************************************************************* * I believe that I made a significant original contribution * * (in my book) by formulating the equation X_{n+1} = P(X_n) as * * a pullback along the structure map of a well founded coalegbra. * ********************************************************************* The example of the gluing construction illustrates the difficulty caused by treating replacment as a footnote to a more elementary theory. There is, as you say, no foundational difficulty in constructing the comma category arising from a functor U:AA-->SS. Although I have copied most of what I have to say about replacement in Section 9.6 of the book in these postings, I am not going to do so for my account of the gluing construction, as it is mathematically too complicated to do so. You will have to get the book itself and read section 7.7. The foundational issue about this construction is its application to consistency issues of various theories. In these, SS is a "semantic" model of the theory in question, maybe the universe "Set" in which we claim to live, and AA is the "syntactic" or "term" model. We have to be careful about calling AA the "free" or "initial" model, as this is exactly the foundational point. The gluing construction is the comma category (SS,U) whose objects are SS-maps of the form X-->U(Gamma), where Gamma is an object of AA and X one of SS. I use the letter Gamma because it is typically a context of the theory under study. One can show that (U,SS) typically inherits the structure of this theory, and pi_1:(SS,U)-->AA preserves it. Since (U,SS) is a model of the theory, whilst AA is the term model, there ought to be an interpretation functor [[-]]:AA-->(U,SS). We have no difficulty in saying what such a functor would be in substance, since we can express it as a fibration EE-->AA of small categories. However, this illustates the difficulty with the word "definable" - if you already had this fibration then there would be nothing left for Replacment to do. The problem is whether the functor or fibration exists. Now, as AA is the term model, we can use recursion over its well founded system of types, terms and proofs. For example, if we already know [[Gamma]] and [[Delta]], we can form the interpretation of the arrow type Gamma -> Delta as the exponential [[ Gamma -> Delta ]] = [[Delta]] ^ [[Gamma]]. However, this is not a valid form of recursion, since recursion defines new TERMS of pre-existing types. We want to form new TYPES. This involves the axiom scheme of replacement. ****************************************************************** * Again, I claim an original contribution in my book in * * recognising that this application of the gluing construction * * to logic requires the axiom scheme of replacement. * ****************************************************************** Notice that replacement is not only a scheme indexed by types, it is also parametric in the theory under study: each theory has its own replacement scheme. If the original theory was algebra, we can characterise the corresponding notion of replacement as what are variously known as dependent products, partial products or W-types. But also, the operation of formulating replacement turns one theory into another, so it can be iterated. Ten years ago, I briefly believed that this leads to inconsistency in ZF. I suspect that this is a necessary though not sufficient "rite of passage" - ie that, only after temporarily believing that it is inconsistent, can one claim to understand what the Axiom of Replacement says. Paul Taylor From rrosebru@mta.ca Wed Mar 19 19:42:25 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 19 Mar 2008 19:42:25 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Jc6pa-0002ac-9e for categories-list@mta.ca; Wed, 19 Mar 2008 19:32:22 -0300 Date: Wed, 19 Mar 2008 13:43:23 -0500 (EST) From: Michael Barr To: Categories list Subject: categories: Re: A question on adjoints MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 105 Actually, F isn't even a functor. The unique arrow 0 --> Ub has to give a canonical arrow F0 = 1 --> b, which there isn't. You could choose one, of course, but it could not be functorial. Actually, I realized the answer to my question cannot be yes. Here's why. Let A be some complete category to be specified later. Let d be a fixed object of A. Let B be set\op and Fa = Hom(a,d). The right adjoint is given by b |---> d^b. It is not entirely trivial to show this, but if my answer were "yes", then you could show that the class of objects that were equalizers of powers of d would be complete. It is obviously closed under products but, over 40 years ago, Isbell gave an example in which it was not closed under equalizers. This much is true: if there is an equalizer of the form a --> UFa ===> Ub, then a ---> UFa ===> UFUFa is an equalizer. Michael On Tue, 18 Mar 2008, Vaughan Pratt wrote: > Isn't the following a counterexample? > > Let A = Set and let B = A\{0} (the category of nonempty sets). Let F send > the empty set in A to the singleton set in B, and otherwise let F and U be > the evident identity functors between A and B. Similarly let \eta and > \epsilon be the identity natural transformations, except for \eta_0 which can > only be the unique function from 0 to 1. Naturality of \eta and \epsilon > depends on both being the identity, except for \eta_0 but that's from the > initial object so all its diagrams commute. > > Then 0 equalizes the two arrows from U1 to U2 but \eta_0 does not equalize > UF\eta a and \eta UFa since the latter two are both 1_1 in A whence they are > equalized by 1. > > Vaughan > > Michael Barr wrote: >> I guess I am getting old and dumb. This question should have been a snap >> for me years ago. It is old fashioned, only a 1-categorical question and >> not about internal vs. external. >> >> Suppose F: A --> B is left adjoint to U: B --> A. Suppose a is an object >> of A and b, b' objects of B such that there is an equalizer >> a ---> Ub ===> Ub'. (The two arrows Ub to UB' are not assumed to be U >> of arrows from B.) Does it follow that a ---> UFa ===> UFUFa is an >> equalizer? The arrows are \eta a, UF\eta a and \eta UFa of course. >> >> Michael >> >> > > From rrosebru@mta.ca Thu Mar 20 11:16:51 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 20 Mar 2008 11:16:51 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JcLUI-0004tm-7j for categories-list@mta.ca; Thu, 20 Mar 2008 11:11:22 -0300 Date: Wed, 19 Mar 2008 16:41:07 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: Categories list Subject: categories: Re: A question on adjoints Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 106 So it *is* a counterexample---to the notion that any old graph theorist can do category theory. Focusing on the naturality, I forgot about functoriality (done that before). More embarrassing is not thinking to perform the easiest test of all category theory, F(0) = 0. And most embarrassing is thinking that Mike could have overlooked such an easy example. Sorry, Mike! Vaughan Michael Barr wrote: > Actually, F isn't even a functor. The unique arrow 0 --> Ub has to give > a canonical arrow F0 = 1 --> b, which there isn't. From rrosebru@mta.ca Thu Mar 20 11:17:23 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 20 Mar 2008 11:17:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JcLZq-0005zy-D5 for categories-list@mta.ca; Thu, 20 Mar 2008 11:17:06 -0300 Date: Thu, 20 Mar 2008 09:13:07 +0000 From: Reiko Heckel MIME-Version: 1.0 To: categories@mta.ca Subject: categories: [gratra] ICGT 2008 - Call for Papers Content-Type: text/plain; charset=ISO-8859-15; format=flowed Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 107 [Please apologize if you receive multiple copies of this message.] Second Call for Papers ------------------------ 4th International Conference on Graph Transformation (ICGT 2008) Leicester, United Kingdom, September 7 - 13, 2008 ----------------------- The 4th International Conference on Graph Transformation (ICGT 2008) will be held in Leicester (United Kingdom) in the second week of September 2008, along with several satellite events. It continues the line of conferences previously held in Barcelona (Spain) in 2002, Rome (Italy) in 2004, and Natal (Brazil) in 2006 as well as a series of six International Workshops on Graph Transformation with Applications in Computer Science between 1978 and 1998, and alternates with the workshop series on Application of Graph Transformation with Industrial Relevance. The conference takes place under the auspices of EATCS, EASST, and IFIP WG 1.3. Awards will be given by EATCS and EASST for the best theoretical and application-oriented papers. Proceedings are planned with Springer's Lecture Notes in Computer Science series. Scope: Graphs are among the simplest and most universal models for a variety of systems, not just in computer science, but throughout engineering and the life sciences. When systems evolve we are interested in the way they change, to predict, support, or react to their evolution. Graph transformation combines the idea of graphs as a universal modelling paradigm with a rule-based approach to specify evolution. The area is concerned with both the theory of graph transformation and their application to a variety of domains. The conference aims at bringing together researchers and practitioners interested in the foundations and application of graph transformation to a variety of areas. Topics of interest include, but are not limited to * foundations and theory of o General models of graph transformatio o High-level and adhesive replacement systems o Node-, edge-, and hyperedge replacement grammars o Parallel, concurrent, and distributed graph transformations o Term graph rewriting o Hierarchical graphs and decompositions of graphs o Logic expression of graph transformation properties o Graph theoretical properties of graph languages o Geometrical and topological aspects of graph transformation o Automata on graphs and parsing of graph languages o Analysis and verification of graph transformation systems o Structuring and modularization concepts for transformation systems o Graph transformation and Petri nets * application to, languages and tool support for o Software architecture o Workflows and business processes o Software quality and testing o Software evolution o Access control and security models o Aspect-oriented development o Model-driven development, especially model transformations o Domain-specific languages o Implementation of programming languages o Bioinformatics and system biology o Natural computing o Image generation and pattern recognition techniques o Massively parallel computing o Self-adaptive systems and ubiquitous computing o Service-oriented applications and semantic web o Rule- and knowledge-based systems Submitted papers should not exceed fifteen (15) pages using Springer's LNCS format, and should contain original research. Simultaneous submission to other conferences with proceedings or submission of material that has already been published elsewhere is not allowed. Important Dates: Submission of title and abstract: April 10, 2008 Submission of complete paper: April 17, 2008 Notification of acceptance: May 15, 2008 Final version due: June 15, 2008 Main conference: September 10-12, 2008 Conference including satellite events: September 7-13, 2008 The following Satellite Events are planned: - Doctoral Symposium Contact: Andrea Corradini - GCM: Workshop on Graph Computation Models Contact: Mohamed Mosbah - GraBaTs: Graph Transformation Tools Contest Contact: Arend Rensink - Tutorial: Introduction to Graph Transformation Contact: Reiko Heckel - PNGT: Petri Nets and Graph Transformations Contact: Paolo Baldan - NCTG: Natural Computing and Graph Transformation Contact: Grzegorz Rozenberg , Ian Petre Venue: Located in the heart of England, Leicester is a truly multi-cultural city. The city is a historic meeting place, where for centuries people of different races and cultures have gathered, creating a rich and unique heritage. This diversity continues today with a thriving ethnic minority community accounting for more than a third of Leicester's population. ICGT 2008 will be held at the University of Leicester's conference facility next to the Universiy's botanic gardens. Organisation Program Chairs Reiko Heckel , University of Leicester, United Kingdom Gabriele Taentzer , Philipps-Universit=E4t Marburg, Germany Local Organisation Reiko Heckel , University of Leicester, United Kingdom Publicity Chair: Karsten Ehrig , University of Leicester, United Kingdom Workshop Chair: D=E9nes Bisztray , University of Leicester, United Kingdom PC members: # Paolo Baldan, Padova (Italy) # Luciano Baresi, Milano (Italy) # Michel Bauderon, Bordeaux (France) # Andrea Corradini, Pisa (Italy) # Hartmut Ehrig, Berlin (Germany) # Gregor Engels, Paderborn (Germany) # Annegret Habel, Oldenburg (Germany) # Reiko Heckel (co-chair), Leicester (UK) # Dirk Janssens, Antwerp (Belgium) # Garbor Karsai, Nashville (USA) # Barbara Koenig, Stuttgart (Germany) # Hans-J=F6rg Kreowski, Bremen (Germany) # Juan de Lara, Madrid (Spain) # Tom Mens, Mons (Belgium) # Mark Minas, M=FCnchen (Germany) # Ugo Montanari, Pisa (Italy) # Mohamed Mosbah, Bordeau (France) # Manfred Nagl, Aachen (Germany) # Fernando Orejas, Barcelona (Spain) # Francesco Parisi-Presicce, Rome (Italy) # Mauro Pezz=E8, Milano (Italy) # John Pfaltz, Charlottesville (Virginia, USA) # Rinus Plasmeijer, Nijmegen (The Netherlands) # Detlef Plump, York (UK) # Arend Rensink, Twente (The Netherlands) # Leila Ribeiro, Porto Alegre (Brasil) # Grzegorz Rozenberg, Leiden (The Netherlands) # Andy Sch=FCrr, Darmstadt (Germany) # Gabriele Taentzer (co-chair), Marburg (Germany) # Hans Vangheluwe, Montreal (Canada) # D=E1niel Varr=F3, Budapest (Hungary) # Albert Z=FCndorf, Kassel (Germany) Further Information can be found at: http://www.cs.le.ac.uk/events/icgt20= 08 --=20 ------------------------------------------ Prof. Dr. Gabriele Taentzer Philipps-Universit=E4t Marburg Fachbereich Mathematik und Informatik Hans-Meerwein-Str. D-35032 Marburg Phone: +49-6421-2821532 Email: taentzer@mathematik.uni-marburg.de ------------------------------------------ From rrosebru@mta.ca Thu Mar 20 11:18:40 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 20 Mar 2008 11:18:40 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JcLbC-0006Hk-Me for categories-list@mta.ca; Thu, 20 Mar 2008 11:18:30 -0300 Date: Thu, 20 Mar 2008 11:07:42 +0100 From: MIME-Version: 1.0 To: destinataires inconnus: ; Subject: categories: Fundamenta Informaticae / Special issue / Machines, Computations and Universality Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 108 Call for paper for the special issue of Fundamenta Informaticae on "Machines, Computations and Universality". This call for paper is open to everyone (it is not restricted to the participant of MCU '07). After the success of MCU '07 at Orleans (France), in Sept. 10-13, 2007, a special issue of Fundamenta Informaticae will be published with contributions on the topics of the conference, which include the following: Digital computation (fundamental classical models): Turing machines, register machines, word processing (groups and monoids), other machines. Digital models of computation: cellular automata, other automata, tiling of the plane, polyominoes, snakes, neural networks, molecular computations, Analog and Hybrid Computations: BSS machines, infinite cellular automata, real machines, quantum computing, computable analysis, abstract geometrical computation. In all these settings: frontiers between a decidable halting problem and an undecidable one in the various computational settings minimal universal codes: size of such a code, namely, for Turing machines, register machines, cellular automata, tilings, neural nets, Post systems computation complexity of machines with a decidable halting problem as well as universal machines, connections between decidability under some complexity class and completeness according to this class, self-reproduction and other tasks, universality and decidability in the real field. Submissions will be refereed and here are the dates for the process: submission dead-line: April 10th, 2008 (strict) notification of accetance/rejection : September, 1st, 2008 final version due: December, 1st, 2008 If you already have a published contribution in the proceedings of the conference (LNCS 4664), we draw your attention on the following: your submission must be sustantially different from the paper of LNCS: it must either contain significantly new results or important proofs that could not be included in the LNCS format; we have to strictly apply this rule. Send your submission to the following address: margens@univ-metz.fr It is important that your submission applies FI's format (see FI's site: http://fi.mimuw.edu.pl/) for your contribution to be examined. There is no apriori limit on the number of pages. The format of FI is large and, in principle, 30 pages is a reasonable limit. If you actually need more,please contact us. Jerome Durand-Lose, Maurice Margenstern, co-chairs of MCU '07 From rrosebru@mta.ca Thu Mar 20 11:20:02 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 20 Mar 2008 11:20:02 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JcLcY-0006Xu-2O for categories-list@mta.ca; Thu, 20 Mar 2008 11:19:54 -0300 From: Thomas Streicher Subject: categories: a tentative answer to Paul To: categories@mta.ca Date: Thu, 20 Mar 2008 14:22:19 +0100 (CET) MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 109 Dear Paul, I do have reveiled my meta-theory, namely ZFC together with the axiom that every set appears as element of some Grothendieck universe. This is not a question of belief but of convenience. I haven't found anything in category theory which needs expressivity beyond that. You ask what I mean by the validity of the statement \forall n:N \exists i : P(X_n)^{X_{n+1}} Iso(i) Well, it has to be read in the internal language of a topos augmented with some `macros' form the language of dependent types (here X -> N is some map in the topos and the formula specifies that it's the family P(N)_{n \in N}. Of course, one might take the pains of Kripke-Joyaling this internal statement BUT why should one want to do so. You are quite right in emphasizing that specifying such a family is one thing and its existence is another one. For the latter purpose one needs the eaxiom of replacement though possibly not its full strength. That's the motivation for my considerations in my "Universes in Toposes". Postulating a class of display maps SS with a strongly generic family E -> U (i.e. this map is in SS and all maps can be obtained as pullback of it) and some desired closure properties. Now supposing N \in U and U being closed under powerset P(-) its is a most simple exercise to construct a map f : N -> U with f(0) = N and f(n+1) = P(f(n)). We have just exploited that P restricts to a map U -> U. This is known since around 1970 when Martin-Loef introduced universes. Of course, he now would not consider something impredicative as the power set functor P but he would consider X \mapsto X^X as an operation on U. Apparently, quite a few category people are not that fond of universes in this sense. So one may ask the question to which extent one may express recursively defined families without universes. The answer given in Paul's book is to require initial fixpoints of endofunctors on some category of families. In case of the example above it would be an endofunctor on EE/N where EE is the topos under consideration. (See www.mathematik.tu-darmstadt.de/~streicher/itpowtop.pdf (temporary!) for one possible way of putting it.) Such an external handling of recursive family is certainly possible but I find it inconvenient because I want to construct my recursive families inside the internal language. In mathematics you never work externally but always internally, i.e. in one single unspecified formal system like ZFC or strengthenings of it (see above). "External" is a logician's idea who considers formal systems, their models and relations between them. But when doing mathematics it's not a good idea to "go external". Thomas PS I find interesting you comments about glueing and the necessity of replacement. I haven't come across this because on the meta-level I use a very strong system where I have it anyway. You are right that via glueing you get a consistency proof for HAH (higher order arithmetic). But this a priori doesn't require replacemnt since Zermelo set theory is sufficient for this purpose. From rrosebru@mta.ca Thu Mar 20 22:27:15 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 20 Mar 2008 22:27:15 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JcVqA-0005bP-KX for categories-list@mta.ca; Thu, 20 Mar 2008 22:14:38 -0300 Date: Thu, 20 Mar 2008 16:20:49 +0000 (GMT) From: Bob Coecke To: categories@mta.ca Subject: categories: Submission deadline March 31 for QUANTUM PHYSICS AND LOGIC & DEVELOPMENTS IN COMPUTATIONAL MODELS, Reykjavik, Iceland, July 12-13, 2008 MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 110 ANNOUNCEMENT/CALL FOR CONTRIBUTIONS: --------------------------------------- Joint International Workshop on: QUANTUM PHYSICS AND LOGIC (QPL'08) DEVELOPMENTS IN COMPUTATIONAL MODELS (DCM'08) July 12-13, 2008, Reykjavik, Iceland. http://web.comlab.ox.ac.uk/oucl/work/bob.coecke/DCM_QPL_08.html --------------------------------------- Programme Committee: Howard Barnum (Los Alamos) Dan Browne (University College London) Bob Coecke (Oxford) Program Co-Chair Vincent Danos (Edinburgh) Andreas Doering (Imperial College London) Viv Kendon (Leeds) Annick Lesne (IHS Paris) Ian Mackie (LIX Paris) Prakash Panangaden (McGill) Program Co-Chair Jon Yard (Los Alamos) Invited speakers: Terry Rudolph (Imperial College London) Andreas Winter (Bristol) --------------------------------------- This ICALP 2008 affiliated joint event combines two (established) workshop series: QUANTUM PHYSICS AND LOGIC (QPL'08): This event has as its goal to bring together researchers working on mathematical foundations of quantum computing and the use of logical tools, new structures, formal languages, semantical methods and other computer science methods for the study quantum behaviour in general. Over the past couple of years there has been a growing activity in these foundational approaches together with a renewed interest in the foundations of quantum theory, which complement the more mainstream research in quantum computation. A predecessor of this event, with the same acronym, called Quantum Programming Languages, was held in Ottawa (2003), Turku (2004), Chicago (2005) and Oxford (2006); with the change of name and a new program committee we wish to emphasise the intended much broader scope of this event, aiming to nourish interaction between modern computer science logic, quantum computation and information, and structural foundations for quantum physics. DEVELOPMENTS IN COMPUTATIONAL MODELS (DCM'08): Besides quantum computing, several new models of computation have emerged in the last few years, and many developments of traditional computational models have been proposed with the aim of taking into account the new demands of computer systems users and the new capabilities of computation engines. A new computational model, or a new feature in a traditional one, usually is reflected in new structural paradigms. The aim of this workshop is to bring together researchers who are currently developing new computational models or new features for traditional computational models, in order to foster their interaction, to provide a forum for presenting new ideas and work in progress, and to enable newcomers to learn about current activities in this area. Previous editions in 2005, 2006 and 2007 were also affiliated to ICALP. Dates: - Submission deadline: March 31 - Acceptance/rejection notification: April 21 - Pre-proceedings versions due: June 15 - Workshop: July 12-13 2007 Submission format: Prospective speakers are invited to submit a 2-5 pages abstract which provides sufficient evidence of results of genuine interest and provides sufficient detail to allows the program committee to assess the merits of the work. Submissions of works in progress are encouraged but must be more substantial than a research proposal. We both encourage submissions of original research as well as research submitted elsewhere. Authors of accepted original research contributions will be invited to submit a full paper to a special issue of a journal yet to be decided on. Submissions should be in Postscript or PDF format and should be sent to Bob Coecke by March 31. Receipt of all submissions will be acknowledged by return email. Accepted contributors will be able to publish extended versions of their abstracts in Electronic Notes in Theoretical Computer Science. The workshop enjoys support from: EPSRC Network Semantics of Quantum Computation (EP/E006833/1) EPSRC ARF The Structure of Quantum Information and its Applications to IT (EP/D072786/1) From rrosebru@mta.ca Fri Mar 21 20:38:39 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 21 Mar 2008 20:38:39 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JcqjR-00003L-0o for categories-list@mta.ca; Fri, 21 Mar 2008 20:33:05 -0300 Date: Fri, 21 Mar 2008 15:21:06 +0100 From: Carlos Areces Subject: categories: E. W. Beth Dissertation Prize: 2008 call for submissions Content-type: text/plain To: undisclosed-recipients:; Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 111 E. W. Beth Dissertation Prize: 2008 call for submissions =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D Since 2002, FoLLI (the European Association for Logic, Language, and Information, www.folli.org) awards the E. W. Beth Dissertation Prize to outstanding dissertations in the fields of Logic, Language, and Information. We invite submissions for the best dissertation which resulted in a Ph.D. degree in the year 2007. The dissertations will be judged on technical depth and strength, originality, and impact made in at least two of the three fields of Logic, Language, and Computation. Inter-disciplinarity is an important feature of the theses competing for the E. W. Beth Dissertation Prize. Who qualifies ~~~~~~~~~~~~~ Nominations of candidates are admitted who were awarded a Ph.D. degree in the areas of Logic, Language, or Information between January 1st, 2007 and December 31st, 2007. There is no restriction on the nationality of the candidate or the university where the Ph.D. was granted. After a careful consideration, FoLLI has decided to accept only dissertations written in English. Dissertations produced in 2007 but not written in English or not translated will be allowed for submission, after translation, also with the call next year (for 2008). Respectively, nominations of full English translations of theses originally written in other language than English and defended in 2006 and 2007 will be accepted for consideration this year, too. Prize ~~~~~ The prize consists of: * a certificate * a donation of 2500 euros provided by the E. W. Beth Foundation. * an invitation to submit the thesis (or a revised version of it) to the new series of books in Logic, Language and Information to be published by Springer-Verlag as part of LNCS or LNCS/LNAI. (Further information on this series is available on the FoLLI site) How to submit ~~~~~~~~~~~~~ Only electronic submissions are accepted. The following documents are required: 1. the thesis in pdf or ps format (doc/rtf not accepted); 2. a ten page abstract of the dissertation in ascii or pdf format; 3. a letter of nomination from the thesis supervisor. Self-nominations are not admitted: each nomination must be sponsored by the thesis supervisor. The letter of nomination should concisely describe the scope and significance of the dissertation and state when the degree was officially awarded; 4. two additional letters of support, including at least one letter from a referee not affiliated with the academic institution that awarded the Ph.D. degree. All documents must be submitted electronically to bethaward2008@gmail.com. Hard copy submissions are not admitted. In case of any problems with the email submission or a lack of notification within three working days after submission, nominators should write to goranko@maths.wits.ac.za or policriti@dimi.uniud.it. Important dates ~~~~~~~~~~~~~~~ Deadline for Submissions: April 30th, 2008. Notification of Decision: July 15th, 2008. Committee : * Anne Abeill=C3=A9 (Universit=C3=A9 Paris 7) * Natasha Alechina (University of Nottingham) * Didier Caucal (IGM-CNRS) * Nissim Francez (The Technion, Haifa) * Valentin Goranko (chair) (University of the Witwatersrand, Johannesburg) * Alexander Koller (University of Edinburgh) * Alessandro Lenci (University of Pisa) * Gerald Penn (University of Toronto) * Alberto Policriti (Universit=C3=A0 di Udine) * Rob van der Sandt (University of Nijmegen) * Colin Stirling (University of Edinburgh) From rrosebru@mta.ca Sun Mar 23 11:11:26 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 23 Mar 2008 11:11:26 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JdQe6-0005WW-Nc for categories-list@mta.ca; Sun, 23 Mar 2008 10:53:58 -0300 Date: Sat, 22 Mar 2008 12:09:08 +0100 From: Venanzio Capretta MIME-Version: 1.0 To: categories@mta.ca Subject: categories: MSFP second call for papers Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 112 SECOND CALL FOR PAPERS This is a reminder that the deadline for submission to MSFP is approachin= g. Second Workshop on MATHEMATICALLY STRUCTURED FUNCTIONAL PROGRAMMING 6 July 2008, Reykjavik - Iceland A satellite workshop of ICALP 2008 PRESENTATION The workshop on Mathematically Structured Functional Programming is devoted to the derivation of functionality from structure. It is a celebration of the direct impact of Theoretical Computer Science on programs as we write them today. Modern programming languages, and in particular functional languages, support the direct expression of mathematical structures, equipping programmers with tools of remarkable power and abstraction. Monadic programming in Haskell is the paradigmatic example, but there are many more mathematical insights manifest in programs and in programming language design: Freyd-categories in reactive programming, symbolic differentiation yielding context structures, and comonadic presentations of dataflow, to name but three. This workshop is a forum for researchers who seek to reflect mathematical phenomena in data and control. The first MSFP workshop was held in Kuressaare, Estonia, in July 2006. An associated special issue of the Journal of Functional Programming is in preparation. INVITED SPEAKERS Andrej Bauer, University of Ljubljana Dan Piponi, Industrial Light and Magic SUBMISSIONS Electronic Notes in Theoretical Computer Science have provisionally agreed to publish the proceedings of MSFP 2008. ENTCS require submissions in LaTeX, formatted according to their guidelines (http://www.entcs.org/prelim.html). Papers must report previously unpublished work and not be submitted concurrently to another conference with refereed proceedings. Programme Committee members, barring the co-chairs, may (and indeed are encouraged to) contribute. Accepted papers must be presented at the workshop by one of the authors. There is no specific page limit, but authors should strive for brevity. We are using the EasyChair software to manage submissions. To submit a paper, please log in at: http://www.easychair.org/conferences/?conf=3Dmsfp2008. TIMELINE: Submission of abstracts: 4 April Submission of papers: 11 April Notification: 16 May Final versions due: 13 June Workshop: 6 July For more information about the workshop, go to: http://msfp.org.uk/ Programme Committee * Yves Bertot, INRIA, Sophia-Antipolis, France * Venanzio Capretta (co-chair), Radboud University, Nijmegen, The Netherlands * Jacques Carette, McMaster University, Hamilton, Ontario, Canada * Thierry Coquand, Chalmers University, G=F6teborg, Sweden * Andrzej Filinski, DIKU, University of Copenhagen, Denmark * Jean-Christophe Filli=E2tre, LRI, Universit=E9 Paris Sud, France * Jeremy Gibbons, Oxford University, England * Andy Gill, Galois Inc., Portland, Oregon, USA * Peter Hancock, University of Nottingham, England * Oleg Kiselyov, FNMOC, Monterey, California, USA * Paul Blain Levy, University of Birmingham, England * Andres L=F6h, Utrecht University, The Netherlands * Marino Miculan, Universit=E0 di Udine, Italy * Conor McBride (co-chair), Alta Systems, Northern Ireland * James McKinna, Radboud University, Nijmegen, The Netherlands * Alex Simpson, University of Edinburgh, Scotland * Tarmo Uustalu, Institute of Cybernetics, Tallinn, Estonia From rrosebru@mta.ca Sat Mar 29 12:34:49 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 29 Mar 2008 12:34:49 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Jfcn7-0000r8-Pb for categories-list@mta.ca; Sat, 29 Mar 2008 12:16:21 -0300 Subject: categories: FMCS 2008, Halifax - please register! To: categories@mta.ca (Categories List) Date: Fri, 28 Mar 2008 17:31:48 -0300 (ADT) MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit From: selinger@mathstat.dal.ca (Peter Selinger) Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 113 FMCS 2008 16th Workshop on Foundational Methods in Computer Science Dalhousie University, Halifax, Canada May 30 - June 1, 2008 http://www.mathstat.dal.ca/~selinger/fmcs2008/ SECOND ANNOUNCEMENT: CALL FOR PARTICIPATION IMPORTANT DEADLINES: *** April 15 *** - please register to speak at the workshop. *** April 15 *** - to apply for student travel support * * * Foundational Methods in Computer Science is an annual workshop bringing together researchers in mathematics and computer science with a focus on the application of category theory in computer science. This year's meeting will be hosted in the Department of Mathematics and Statistics at Dalhousie University in Halifax, Canada. There will be an informal welcome reception in the evening of May 29. The scientific program starts on May 30, and consists of a day of tutorials aimed at students and newcomers to category theory, as well as a day and a half of research talks. The meeting ends at mid-day on June 1. FMCS 2008 takes place one week after MFPS 2008 (which will be held at the University of Pennsylvania). We hope that this will enable and encourage participants from overseas to attend both conferences! TUTORIAL LECTURES: There will be four tutorial lectures, presented by: Pieter Hofstra (Ottawa) Ernie Manes (Massachusetts) Paul-Andre Mellies (Paris 7) Andrea Schalk (Manchester) SPECIAL SESSION: There will be a special session in honor of Ernie Manes' 65th birthday. The special session will be organized by Phil Mulry. TO GIVE A TALK: Prospective speakers are requested to send a title and optional abstract to fmcs2008@mathstat.dal.ca by *** April 15 ***. Late submissions may be considered if there is space. All submissions will be acknowledged by return email. STUDENT SUPPORT: Graduate student participation is encouraged at FMCS. Students will pay a reduced registration fee. We will also be able to provide limited support for travel and accommodations to students. If you are interested in this, please send a request to fmcs2008@mathstat.dal.ca by *** April 15 ***. Please also arrange for a letter of reference from your supervisor or appropriate other person. ACCOMMODATIONS: We have reserved a block of rooms at the King's College residences. The rate, including taxes, are $37.37 per night for a single room, and $56.04 for a double room. Reservations can be made by sending an e-mail to conferences@admin.ukings.ns.ca and mentioning "FMCS 2008". A reservation form is available from the workshop website. It is preferred that you make your reservations by *** April 15 *** to ensure availability. For those wishing to stay in a hotel or bed & breakfast, some information is available on the conference website. REGISTRATION: Please register for the meeting by emailing fmcs2008@mathstat.dal.ca, preferably by *** April 15 ***. There will be an on-site registration fee of $120 to cover meeting costs. A discounted registration fee is available for students and for researchers without grant. MAPS AND LOCAL INFORMATION: Local information, including maps, is available from the conference website, http://www.mathstat.dal.ca/~selinger/fmcs2008/ PREVIOUS MEETINGS: Previous FMCS meetings were held in Pullman (1992), Portland (1993), Vancouver (1994), Kananaskis (1995), Pullman (1996), Portland (1998), Kananaskis (1999), Vancouver (2000), Spokane (2001), Hamilton (2002), Ottawa (2003), Kananaskis (2004), Vancouver (2005), Kananaskis (2006), and Hamilton (2007). ORGANIZING COMMITTEE: Robin Cockett (Calgary) John MacDonald (UBC) Phil Mulry (Colgate) Dorette Pronk (Dalhousie) Robert Seely (McGill) Peter Selinger (Dalhousie) LOCAL ORGANIZERS: Dorette Pronk (Dalhousie) Peter Selinger (Dalhousie) * From rrosebru@mta.ca Mon Mar 31 21:52:29 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 31 Mar 2008 21:52:29 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JgUVu-0001e1-A7 for categories-list@mta.ca; Mon, 31 Mar 2008 21:38:10 -0300 Date: Mon, 31 Mar 2008 15:51:39 -0400 (EDT) Subject: categories: exploiting similarities and analogies From: "Al Vilcius" To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 114 Dear Categorists, Has anyone explored, either formally or informally, the connection betwee= n the Melzak Bypass Principle (MBP) and adjoints? The MBP (aka "the conjugacy principle" which embraces and generalizes Jacobi inversion) Ref MR696771 =20 http://www.ams.org/mathscinet/pdf/696771.pdf (and no, it does not appear in either Wikipedia or PlanetMath, yet) is somewhat heuristic in character, suggesting : Transform the problem (T), Solve(S), Transform back(T^1), as a "bypass" given by (T^1)ST, which looks like conjugation. Melzak himself refers to adjoints (quite tangentially) as "being bypasses= , though dressed up and served forth exotically" p.106 ibid. (I do recall that adjoints were generally seen as pretty exotic in the early 1970's when I was a graduate student at UBC, to my great chagrin). The MBP is acclaimed in MM vol.57 No.3 May 1984 as "a device for exploring analogies", or as "a dazzling attempt to comprehend complexity". Perhaps "bypass" could also be seen in the words of W.W. Tait (1996) "the propositions about the abstract objects translate into propositions about the things from which they are abstracted and, in particular, the truth o= f the former is founded on the truth of the latter". http://home.uchicago.edu/~wwtx/frege.cantor.dedekind.pdf . For me, "bypass" is a kenning for quasi-inverse or ad joint pairs. Motivation for seeking such a connection is not really to revive a 25 or 30 year old idea (as brilliant as Melzak's insights were, of course), but rather "to facilitate invention and discovery" (in Melzak's own words), and to find additional (as well as interdisciplinary) sources of instance= s of adjoints, possibly as a way to make adjoints more immediately relevant in any introductory discussion of categories, since, of course, adjoints are undoubtedly one of the most successful concepts within category theory. Further inspiration could be found in the Brown/Porter "Analogy" paper http://www.bangor.ac.uk/~mas010/eureka-meth1.pdf and others, along with a passion for invention and discovery through the continued pursuit of Unity and Identity of Opposites (UIO) - obviously referring to Bill Lawvere. Furthermore, I would also like to see this connection developed for practical reasons, applied to various situations, in particular to the structure of the www as an anthropomorphic creation that could benefi= t from further categorical perspective, given by the learned categorists I respect the most. I look forward to your thoughts and comments. ..... Al Al Vilcius Campbellville, ON, Canada