From MAILER-DAEMON Sun Sep 7 20:02:16 2008 Date: 07 Sep 2008 20:02:16 -0300 From: Mail System Internal Data Subject: DON'T DELETE THIS MESSAGE -- FOLDER INTERNAL DATA Message-ID: <1220828536@mta.ca> X-IMAP: 1217984894 0000000069 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 Tue Aug 5 22:06:49 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 05 Aug 2008 22:06:49 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KQXU7-0001W3-45 for categories-list@mta.ca; Tue, 05 Aug 2008 22:06:39 -0300 Date: Sat, 02 Aug 2008 17:01:37 -0700 From: posina@salk.edu To: categories@mta.ca Subject: categories: place-value notation MIME-Version: 1.0 Content-Type: text/plain;charset=ISO-8859-1;DelSp="Yes";format="flowed" Content-Disposition: inline Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 1 Hi, Category theoretic notions such as sum and colimit can be thought of as generalizations of the elementary arithmetic operation of adding numbers. Given that place-value notation is the syntax (in the sense of underlying structure or format) of addition, is there a category theoretic account of place-value notation? thank you, posina From rrosebru@mta.ca Wed Aug 6 15:48:23 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 06 Aug 2008 15:48:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KQo25-0000si-8n for categories-list@mta.ca; Wed, 06 Aug 2008 15:46:49 -0300 Date: Wed, 6 Aug 2008 16:00:56 +0100 (BST) From: "Prof. Peter Johnstone" To: categories@mta.ca Subject: categories: Re: Paper on slice stability in Locale Theory 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: 2 On Thu, 31 Jul 2008, Townsend, Christopher wrote: > Feedback is welcome. For example, I have always attributed the result > Loc/Y =3D Loc_Sh(Y) to Joyal and Tierney. Am I right?=20 > I think it predates the Joyal--Tierney work by a couple of years. It's (more or less) present in the long Fourman--Scott paper on sheaves and logic in SLN 753 (the Proceedings of the 1977 Durham symposium), but they don't claim originality for it. Peter Johnstone From rrosebru@mta.ca Thu Aug 7 21:25:11 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 07 Aug 2008 21:25:11 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KRFke-0005Zz-4q for categories-list@mta.ca; Thu, 07 Aug 2008 21:22:40 -0300 From: peasthope@shaw.ca Date: Thu, 7 Aug 2008 11:27:16 +0000 Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 To: categories@mta.ca Subject: categories: Category 2 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 3 Folk, The title is revised and notations "top" and "bottom"=20 have been added. The latest scan is here. http://carnot.yi.org/Category2.jpg Regards, ... Peter E. --=20 http://members.shaw.ca/peasthope/ http://carnot.yi.org/ =3D http://carnot.pathology.ubc.ca/ From rrosebru@mta.ca Fri Aug 8 20:53:13 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 08 Aug 2008 20:53:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KRbjb-00067m-BO for categories-list@mta.ca; Fri, 08 Aug 2008 20:51:03 -0300 To: categories@mta.ca From: Michael Fourman Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit Mime-Version: 1.0 (Apple Message framework v926) Subject: categories: Re: Paper on slice stability in Locale Theory Date: Fri, 8 Aug 2008 21:40:33 +0100 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 4 The result was in the air - and I'm sure that Joyal in particular saw it clearly. For me it (or at least the logical version of it given in Fourman- Scott) came as an answer to Dana's request for a description of an internal cHa in a category of "Omega sets". I don't think I had already heard it from Joyal - but I'm sure he already knew it. It was easy to see that an object of Loc/Omega represented an internal cHa, by generalisation of the representation of examples such as the internal cHa O(R) in Sh(X) as O(RxX) -> O(X), or relaxation of the representation of sheaves by local homeomorphisms. Our paper was deliberately (and perhaps mistakenly) concrete rather than abstract, so we didn't think to phrase the representation explicitly as an equivalence of categories. A related (but even more logical and obstinately pointed) result is given in Michael P. Fourman. T1 spaces over topological sites. J. Pure and Applied Algebra, 27(3):223-224, March 1983. On 6 Aug 2008, at 16:00, Prof. Peter Johnstone wrote: > On Thu, 31 Jul 2008, Townsend, Christopher wrote: > >> Feedback is welcome. For example, I have always attributed the result >> Loc/Y =3D Loc_Sh(Y) to Joyal and Tierney. Am I right?=20 >> > I think it predates the Joyal--Tierney work by a couple of years. > It's (more or less) present in the long Fourman--Scott paper on > sheaves and logic in SLN 753 (the Proceedings of the 1977 Durham > symposium), but they don't claim originality for it. > > Peter Johnstone > > > > _______________________________________________ > categories mailing list > categories@inf.ed.ac.uk > http://lists.inf.ed.ac.uk/mailman/listinfo/categories > > Professor Michael Fourman FBCS CITP Head of Informatics University of Edinburgh Informatics Forum 10 Crichton Street Edinburgh EH8 9AB Tel: +44 131 650 2690 michael.fourmaned.ac.uk From rrosebru@mta.ca Tue Aug 12 21:34:54 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 12 Aug 2008 21:34:54 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KT4Hc-0002TW-1i for categories-list@mta.ca; Tue, 12 Aug 2008 21:32:12 -0300 Subject: categories: Set as a monoidal category To: categories@mta.ca (Categories List) Date: Tue, 12 Aug 2008 21:23:59 -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: 5 Dear Categoreans, I know three monoidal structures on the category of sets, all of them symmetric. Two are the product and coproduct, and I'll leave it to your imagination to figure out the third one. My question is: are these the only three? Proofs, counterexamples, or references appreciated. Thanks, -- Peter From rrosebru@mta.ca Wed Aug 13 10:19:07 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 13 Aug 2008 10:19:07 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KTGEt-0001ji-NF for categories-list@mta.ca; Wed, 13 Aug 2008 10:18:11 -0300 Date: Tue, 12 Aug 2008 20:45:38 -0700 From: Toby Bartels To: Categories List Subject: categories: Co-categories 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: 6 I've been thinking idly about a concept dual to categories in much the same way that co-algebras are dual to algebras, and I've decided that I'd like to more about it. To be precise, if V is a monoidal category, then a category enriched over V has maps [A,B] (x) [B,C] -> [A,C], while a cocategory enriched over V has maps [A,C] -> [A,B] (x) [B,C]. (You can fill in the rest of the definition for yourself.) Searching Google, this concept appears to be known (under this name) in the case where V is Abelian, but I'm not so interested in that. I'm more interested in the case where V is a pretopos (like Set) equipped with the coproduct (disjoint union) as the monoidal structure (x). My motivation is that this concept is important in constructive analysis when V is a Heyting algebra equipped with disjunction as (x). (This defines a V-valued apartness relation on the set of objects; but I'm stating even this fact in more generality than I've ever seen.) So if anyone has heard of this concept where V is not assumed abelian, or even knows of a good introduction where V is assumed abelian, then I would be interested in references. --Toby From rrosebru@mta.ca Wed Aug 13 10:19:07 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 13 Aug 2008 10:19:07 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KTGFa-0001nZ-R3 for categories-list@mta.ca; Wed, 13 Aug 2008 10:18:54 -0300 Date: Wed, 13 Aug 2008 09:36:36 +0000 (GMT) From: RONALD BROWN Subject: categories: Re: Set as a monoidal category To: Categories List MIME-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 7 Dear Categoreans, =0A=0AIt is standard practice to answer a different quest= ion! But I can't resist referring to =0AR. Brown ``Ten topologies for $X\= times Y$'', {\em Quart. J.Math.}=0A(2) 14 (1963), 303-319.=0Aand asking if= one can modify these topologies or underlying sets by some process to work= sensibly for the category of sets? (a compact subset of a discrete space i= s of course finite). Maybe it is not possible. =0A=0ARonnie=0A=0A=0A=0A=0A= =0A=0A----- Original Message ----=0AFrom: Peter Selinger =0ATo: Categories List =0ASent: Wednesday, 13 Au= gust, 2008 1:23:59 AM=0ASubject: categories: Set as a monoidal category=0A= =0ADear Categoreans,=0A=0AI know three monoidal structures on the category = of sets, all of them=0Asymmetric. Two are the product and coproduct, and I'= ll leave it to=0Ayour imagination to figure out the third one.=0A=0AMy ques= tion is: are these the only three? Proofs, counterexamples, or=0Areferences= appreciated.=0A=0AThanks, -- Peter From rrosebru@mta.ca Wed Aug 13 10:19:07 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 13 Aug 2008 10:19:07 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KTGEJ-0001gU-IS for categories-list@mta.ca; Wed, 13 Aug 2008 10:17:35 -0300 Content-class: urn:content-classes:message MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Subject: categories: RE: Set as a monoidal category Date: Wed, 13 Aug 2008 11:36:34 +1000 From: "Stephen Lack" To: "Categories List" Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 8 Dear Peter, There's a paper=20 Algebraic categories with few monoidal biclosed structures or none of Foltz, Lair, and Kelly which studies monoidal closed structures on = various categories, and shows that the cartesian closed one is the only = possibility for Set. More generally, it shows that for many categories we know well, the only = possible monoidal closed structures are the ones we know well. But this depends heavily on the closedness. Without that, as you say, = one can use the cocartesian monoidal structure (the coproduct).=20 Here's a further infinite family of monoidal structures on Set. Let A be = any set. Then=20 define the tensor product * by X*Y=3DAXY+X+Y. Steve. -----Original Message----- From: cat-dist@mta.ca on behalf of Peter Selinger Sent: Wed 8/13/2008 10:23 AM To: Categories List Subject: categories: Set as a monoidal category =20 Dear Categoreans, I know three monoidal structures on the category of sets, all of them symmetric. Two are the product and coproduct, and I'll leave it to your imagination to figure out the third one. My question is: are these the only three? Proofs, counterexamples, or references appreciated. Thanks, -- Peter From rrosebru@mta.ca Wed Aug 13 20:02:52 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 13 Aug 2008 20:02:52 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KTPLQ-0000GF-US for categories-list@mta.ca; Wed, 13 Aug 2008 20:01:32 -0300 From: MartDowd@aol.com Date: Wed, 13 Aug 2008 10:37:42 EDT Subject: categories: advanced undergraduate algebra text 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: 9 My self-published advanced undergraduate algebra text "Introductory Algebra, Topology, and Category Theory" has been available by mail order since mid-2006. Recently I have made it available at "_www.amazon.com_ (http://www.amazon.com) ". Also, it has received a favorable review from the MAA online reviews ("_www.maa.org/reviews_ (http://www.maa.org/reviews) "). This text is currently the only one available which covers such a wide range of material in a single volume. It is ideally suited for a number of uses, including as a supplementary text for any undergraduate algebra course. The MAA review recommends it as a second text in algebra for students interested in mathematics. The table of contents and first chapter may be viewed at "_www.hyperonsoft.com_ (http://www.hyperonsoft.com) ". Martin Dowd From rrosebru@mta.ca Wed Aug 13 20:02:52 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 13 Aug 2008 20:02:52 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KTPMF-0000JH-4S for categories-list@mta.ca; Wed, 13 Aug 2008 20:02:23 -0300 Date: Wed, 13 Aug 2008 17:29:23 +0100 (BST) From: Richard Garner To: Categories List Subject: categories: Re: Co-categories 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: 10 Dear Toby, What you call a cocategory enriched over V can also be described as a category enriched over V^op. These have been studied by Paddy McCrudden in his thesis under the name "coalgebroids" (= many-object coalgebras). The main results are on a generalised notion of Tannakian duality; and on transfer of extra structure across this duality. See respectively: [1] Paddy McCrudden, Categories of Representations of Coalgebroids, Advances in Mathematics Volume 154, Issue 2, Pages 299-332 [2] Paddy McCrudden, Balanced Coalgebroids Theory and Applications of Categories, Vol. 7, pp 71-147. Richard --On 12 August 2008 20:45 Toby Bartels wrote: > I've been thinking idly about a concept dual to categories > in much the same way that co-algebras are dual to algebras, > and I've decided that I'd like to more about it. > To be precise, if V is a monoidal category, > then a category enriched over V has maps [A,B] (x) [B,C] -> [A,C], > while a cocategory enriched over V has maps [A,C] -> [A,B] (x) [B,C]. > (You can fill in the rest of the definition for yourself.) > > Searching Google, this concept appears to be known (under this name) > in the case where V is Abelian, but I'm not so interested in that. > I'm more interested in the case where V is a pretopos (like Set) > equipped with the coproduct (disjoint union) as the monoidal structure (x). > My motivation is that this concept is important in constructive analysis > when V is a Heyting algebra equipped with disjunction as (x). > (This defines a V-valued apartness relation on the set of objects; > but I'm stating even this fact in more generality than I've ever seen.) > > So if anyone has heard of this concept where V is not assumed abelian, > or even knows of a good introduction where V is assumed abelian, > then I would be interested in references. > > > --Toby > > > From rrosebru@mta.ca Wed Aug 13 20:03:09 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 13 Aug 2008 20:03:09 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KTPMk-0000Ku-4t for categories-list@mta.ca; Wed, 13 Aug 2008 20:02:54 -0300 Date: Wed, 13 Aug 2008 22:13:00 +0100 (BST) From: Bob Coecke To: categories@mta.ca Subject: categories: Program of Categories, Logic and Foundations of Physics, August 23-24, Oxford 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: 11 The program of the next workshop on Categories, Logic and Foundations of Physics, which will take place at Oxford University, August 23-24, is now available: SATURDAY ---------------- 11:00 - 12:30 TUTORIAL I: QUANTUM FORMALISM (B. Coecke) Von Neumann axioms, Dirac calc., mixed operations, Gleason's and Wigner's thm. 12:30 - 14:00 LUNCH 14:00 - 15:30 TUTORIAL II: CONCEPTUAL ISSUES (A. Doering) Measurement problem, Bell's thm, Kochen-Specker(-Conway) thm, GHZ argument. 15:30 - 16:00 BREAK 16:00 - 17:00 KEITH HANNABUS (Oxford - Mathematics) Categories and non-associative C*-algebras in quantum field theory 17:00 - 18:00 (Kansas State - Mathematics) Model Categories in quantum gravity SUNDAY ------------- 10:00 - 11:00 JOHN BARRETT (Nottingham - Mathematics) Knots and links in braided quantum field theory 11:00 - 11:15 BREAK 11:15 - 12:00 SIMON PERDRIX (Oxford - Computing) TBA 12:00 - 12:45 MEHRNOOSH SADRZADEH (Paris VII - Computing) What is the vector space content of what we say? ... a categorical approach to distributed meaning. 12:45 - 14:00 LUNCH 14:00 - 14:50 STEVE VICKERS (Birmingham - Computing) TBA 14:50 - 15:40 CHRIS FEWSTER (York - Mathematics) Categories in QFT in curved spacetime 15:40 - 16:10 BREAK 16:10 - 17:00 SIMON WILLERTON (Sheffield - Maths; YouTube Catsters) TBA Travel information, location and accommodation information is at: * http://www.comlab.ox.ac.uk/people/bob.coecke/CLOP_info.html Programs and videos of previous workshops and more information on the series are available from: * http://categorieslogicphysics.wikidot.com/ From rrosebru@mta.ca Thu Aug 14 08:58:42 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 14 Aug 2008 08:58:42 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KTbQA-0006Zd-PT for categories-list@mta.ca; Thu, 14 Aug 2008 08:55:14 -0300 Subject: categories: Re: Set as a monoidal category To: categories@mta.ca (Categories List) Date: Wed, 13 Aug 2008 23:28:39 -0300 (ADT) 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: 12 Peter Selinger wrote yesterday: > > I know three monoidal structures on the category of sets, all of them > symmetric. Two are the product and coproduct, and I'll leave it to > your imagination to figure out the third one. > > My question is: are these the only three? Proofs, counterexamples, or > references appreciated. Several people took up the challenge, and provided me with interesting monoidal structures on Set. I'll briefly summarize the correct responses that I have received. Let me begin by revealing the "third" monoidal structure that I had in mind. It is X*Y = XY + X + Y. (1) It is easy to see that this satisfies the axioms if one rewrites it, by an abuse of notation, as X*Y = (X+1)(Y+1) - 1. Steve Lack pointed out that this is part of an infinite family of monoidal structures, each defined by X*Y = XSY + X + Y, (2) where S is some fixed set. Of course, for S=0 this is just the coproduct, and for S=1, this is the same as (1). In the case |S| > 1, coherence is not totally obvious; in fact, there are two possible natural isomorphisms (X*Y)*Z -> X*(Y*Z), deriving from the two natural maps from S^2 to itself. Only one of them is coherent. The coherence proof is somewhat simpler if one writes (2) in the form I have given, rather than in Steve's original form SXY + X + Y. It is interesting to note that, contrary to appearances, Steve's monoidal structure is not symmetric (not even braided) for |S| > 1. There is only one candidate braiding map X*Y -> Y*X, and it fails to satisfy the hexagon axiom. Ralph Loader proposed another monoidal structure, not contained in Steve's family: let X*Y be the set of non-empty finite sequences in X+Y, with no two consecutive elements from the same component of X+Y. Using the Kleene star, this can be written as X*Y = (XY)^* (X+XY) + (YX)^* (Y+YX). (3) (Here, A^* is the list monad, i.e., the initial solution for A^* = 1 + A A^*, also known as the Kleene star). After Ralph saw Steve's family, he noticed that his construction can also be generalized to an infinite family, by alternating list elements with elements of S, namely: X*Y = (XSYS)^* (X+XSY) + (YSXS)^* (Y+YSX). (4) The case S=0 is again the coproduct, and the case S=1 is of course (3). Unlike Steve's family, these tensors appear to be symmetric for all S. Also unlike Steve's family, the construction does not restrict to the category of finite sets. Jeff Egger contributed another symmetric monoidal structure, which he calls "par", and which is defined by: X*Y = X if Y=0 X*Y = Y if X=0 (5) X*Y = 1 if both X and Y are non-empty. This can be uniquely extended to morphisms such that f*0 = f = 0*f. It is perhaps interesting to note that an attempt to make Jeff's construction into an infinite family, by replacing "1" by some pointed set S, *almost* succeeds: the resulting operation is functorial, with coherent associativity and unit isomorphism. The only problem is that associativity fails to be a natural transformation. I mention it here because it might make for a really neat exercise in a course. In summary, we have two infinite families (2) and (4) (both including coproduct), plus product and Jeff's "par" (5). I admit that I did not expect so many non-trivial monoidal structures to exist on Set, and I now expect that there are many more. A complete classification would be interesting, but is perhaps too much to expect. I will close with another challenge: consider only *symmetric* monoidal structures on the category of *finite* sets. So far, we have seen four such structures, namely product, coproduct, and the tensors (1) and (5). Are these the only four? -- Peter P.S. monoidal *closed* structures, as mentioned in Steve's message, are an entirely different ball game. The requirement that tensor is left adjoint implies that it preserves colimits in each components; on Set, everything is therefore determined by 1*1. The only possibility is 1*1=1, which yields the usual cartesian-closed structure. From rrosebru@mta.ca Thu Aug 14 14:06:18 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 14 Aug 2008 14:06:18 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KTgFw-0007hi-4U for categories-list@mta.ca; Thu, 14 Aug 2008 14:05:00 -0300 Date: Thu, 14 Aug 2008 16:29:57 +0200 From: Chris Heunen MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Sheaves in Geometry and Quantum theory, a workshop in Nijmegen 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: 13 Dear all, This is to announce a three-day workshop on the interplay between logic, topology/geometry and quantum theory: Sheaves in Geometry and Quantum theory September 3-5, 2008 Radboud University Nijmegen, the Netherlands funded by the Fellowship of Geometry and Quantum Theory (GQT-cluster) and Stichting Compositio Mathematica. For more information, see the website: http://www.cs.ru.nl/~heunen/lgqt Invited (confirmed) speakers are: * Andreas Doering (Imperial College London) * Anders Kock (Aarhus) * Jaap van Oosten (Utrecht) * Erik Palmgren (Uppsala) * Pedro Resende (Lisbon) * Isar Stubbe (Antwerp) * Steve Vickers (Birmingham) Registration is free, but for logistic reasons, please inform Chris Heunen (heunen@math.ru.nl) if you plan to attend. Best wishes, Mai Gehrke, Chris Heunen, Klaas Landsman, Ieke Moerdijk, and Bas Spitters From rrosebru@mta.ca Sun Aug 17 17:30:57 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 17 Aug 2008 17:30:57 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KUorZ-0007gN-Fa for categories-list@mta.ca; Sun, 17 Aug 2008 17:28:33 -0300 Date: Sun, 17 Aug 2008 14:57:37 -0400 (EDT) From: Michael Barr To: Categories list Subject: categories: Term used in spectral sequences MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 14 As some of you know, my wife and I are translating Grothendieck's Tohoku paper. Originally, it was suggested that it be published as a TAC reprint, but Grothendieck refuses permission because he "does not believe in copyright". So I thought to retype it and post it on my own site (so sue). I then realized that translation would be easier than retyping. Which brings me to a translation problem. I am not expert in spectral sequences and what I know is from Cartan-Eilenberg. I cannot related G's definition to theirs. G defines a spectral sequence as a pair E=(E^{p,q}_r,E^n), all indexed objects of an abelian category subject to five conditions. The first three refer only to the E^{p,q}_r, including that for each pair p,q the E^{p,q}_r stabilize vis-vis r to a term he calls E^{p,q}_\infty (and not E^{p,q}). The fourth assumes "isomorphisms $\beta^{p,q}:E^{pq}\to G^p(E^{p+q})$. The family $(E^n)$ without filtrations is called the \emph{l'aboutissement} of the spectral sequence $E$." The E^n are assumed filtered and G^p is the associated grading: G^p(A)=F^p(A)/F^{p-1}(A). Now this makes no sense. The only thing called E^{pq} would be the term E^n for n=pq and this is really unlikely. I strongly suspect the domain of \beta^{p,q} is intended to be E^{p,q}_\infty. Finally does anyone have any idea how "aboutissement" is to be translated. It means something like limit, but the usual term for that is of course "limite". The Cartan-Eilenberg development is different enough that there seems not to be any corresponding word. Michael From rrosebru@mta.ca Sun Aug 17 17:30:57 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 17 Aug 2008 17:30:57 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KUosB-0007ip-7X for categories-list@mta.ca; Sun, 17 Aug 2008 17:29:11 -0300 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: KT Chen's smooth CCC, a correction Date: Sun, 17 Aug 2008 15:07:56 -0400 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 15 In my review of Anders Kock's Synthetic Differential Geometry, Second Edition, there is a wrong statement that I want to correct. (This was in the SIAM REVIEW, vol. 49, No.2 pp 349-350). The statement was that Chen's category does not include the representability of smooth function spaces. But from his paper In Springer Lecture Notes in Mathematics,vol 1174, pp 38-42 it is clear that it does. I thank Anders=20 for pointing out this slip. This is a good opportunity to emphasize that the works of KT Chen and of Alfred Frolicher (that were referred to in the beginning of the above review) contain several contributions of value both to applications and to more topos-theoretic formulations. For example,=20 Frolicher's use of Lemmas by Boman and others reveals how little of the specific parameter "smooth" needs to be given to the very general machinery of adjoint functors and abstact sets in order to obtain smooth infinite dimensional spaces of all kinds. (Namely a suitable topos of actions by only unary operations on the line is fully embedded in the desired topos in such a way that the algebraic theory of n-ary operations that naturally exist in=20 the small one determines the whole algebraic category whose sheaves include the large one.) And Chen's smooth space of piecewise-smooth curves can surely be further applied, as can his=20 special use of convex models for plots. Bill Lawvere From rrosebru@mta.ca Mon Aug 18 09:26:08 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 18 Aug 2008 09:26:08 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KV3mX-00060e-4Y for categories-list@mta.ca; Mon, 18 Aug 2008 09:24:21 -0300 Date: Sun, 17 Aug 2008 17:51:12 -0400 From: jim stasheff MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: KT Chen's smooth CCC, a correction Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 16 Bill, Happy to see you contributing to the renaissance in interest in Chen's work. It would be good to post your msg to the n-category cafe blog whee there's been an intense discussion of `smooth spaces' i various incarnaitons. jim http://golem.ph.utexas.edu/category/2008/05/convenient_categories_of_smoot.html wlawvere@buffalo.edu wrote: > In my review of Anders Kock's Synthetic > Differential Geometry, Second Edition, > there is a wrong statement that I want to correct. > (This was in the SIAM REVIEW, vol. 49, No.2 > pp 349-350). The statement was that Chen's > category does not include the representability > of smooth function spaces. But from his paper > In Springer Lecture Notes in Mathematics,vol > 1174, pp 38-42 it is clear that it does. I thank Anders > for pointing out this slip. > > This is a good opportunity to emphasize that > the works of KT Chen and of Alfred Frolicher > (that were referred to in the beginning of the > above review) contain several contributions > of value both to applications and to more > topos-theoretic formulations. For example, > Frolicher's use of Lemmas by Boman and others > reveals how little of the specific parameter "smooth" > needs to be given to the very general machinery of > adjoint functors and abstact sets in order to obtain > smooth infinite dimensional spaces of all kinds. > (Namely a suitable topos of actions by only unary > operations on the line is fully embedded > in the desired topos in such a way that the algebraic > theory of n-ary operations that naturally exist in > the small one determines the whole algebraic category whose > sheaves include the large one.) > And Chen's smooth space of piecewise-smooth > curves can surely be further applied, as can his > special use of convex models for plots. > > Bill Lawvere > > > > From rrosebru@mta.ca Mon Aug 18 09:26:08 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 18 Aug 2008 09:26:08 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KV3nB-000646-2S for categories-list@mta.ca; Mon, 18 Aug 2008 09:25:01 -0300 Date: Sun, 17 Aug 2008 20:08:59 -0400 (EDT) From: Michael Barr To: Categories list Subject: categories: abutment 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: 17 That seems to be the translation of the word, thanks to several. That still leaves that mysterious E^{pq} ---> G^p(E^{p+q}) to correct, since I do not believe that it is correct as stands. Michael From rrosebru@mta.ca Mon Aug 18 09:26:26 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 18 Aug 2008 09:26:26 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KV3oN-0006Cr-Nk for categories-list@mta.ca; Mon, 18 Aug 2008 09:26:15 -0300 Date: Sun, 17 Aug 2008 17:16:58 -0700 From: John Baez To: categories Subject: categories: KT Chen's smooth CCC 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: 18 Hi - Bill Lawvere mentioned that KT Chen had a cartesian closed category of smooth spaces. I've found this very useful in my work on geometry. I kept wanting more properties of this category, so finally my student Alex Hoffnung and I wrote a paper about it: Convenient Categories of Smooth Spaces http://arxiv.org/abs/0807.1704 Abstract: A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace or quotient space of a Chen space is a Chen space, and the space of smooth maps between Chen spaces is again a Chen space. Souriau's "diffeological spaces" share these convenient properties. Here we give a unified treatment of both formalisms. Following ideas of Dubuc, we show that Chen spaces, diffeological spaces, and even simplicial complexes are examples of "concrete sheaves on a concrete site". As a result, the categories of such spaces are locally cartesian closed, with all limits, all colimits, and a weak subobject classifier. For the benefit of differential geometers, our treatment explains most of the category theory we use. In particular, at some point we break down and admit we're dealing with a "quasitopos". Best, jb From rrosebru@mta.ca Mon Aug 18 09:27:21 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 18 Aug 2008 09:27:21 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KV3pH-0006L5-Fz for categories-list@mta.ca; Mon, 18 Aug 2008 09:27:11 -0300 Date: Mon, 18 Aug 2008 10:58:47 +0100 (BST) From: "Prof. Peter Johnstone" To: Categories List Subject: categories: Re: Co-categories 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 I was expecting Peter Lumsdaine to reply to this, but perhaps he's away. In discussions with Steve Awodey and myself, Peter recently established the fact that every co-category in a pretopos is a co-equivalence relation; more specifically, the "co-domain" and "co-codomain" maps (sorry, but I can't see any other way to describe them) are the cokernel pair of a (unique) monomorphism (namely, their equalizer). Peter Johnstone On Tue, 12 Aug 2008, Toby Bartels wrote: > I've been thinking idly about a concept dual to categories > in much the same way that co-algebras are dual to algebras, > and I've decided that I'd like to more about it. > To be precise, if V is a monoidal category, > then a category enriched over V has maps [A,B] (x) [B,C] -> [A,C], > while a cocategory enriched over V has maps [A,C] -> [A,B] (x) [B,C]. > (You can fill in the rest of the definition for yourself.) > > Searching Google, this concept appears to be known (under this name) > in the case where V is Abelian, but I'm not so interested in that. > I'm more interested in the case where V is a pretopos (like Set) > equipped with the coproduct (disjoint union) as the monoidal structure (x). > My motivation is that this concept is important in constructive analysis > when V is a Heyting algebra equipped with disjunction as (x). > (This defines a V-valued apartness relation on the set of objects; > but I'm stating even this fact in more generality than I've ever seen.) > > So if anyone has heard of this concept where V is not assumed abelian, > or even knows of a good introduction where V is assumed abelian, > then I would be interested in references. > > > --Toby > > > From rrosebru@mta.ca Mon Aug 18 19:26:22 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 18 Aug 2008 19:26:22 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVD8q-0003yj-1m for categories-list@mta.ca; Mon, 18 Aug 2008 19:24:01 -0300 Date: Mon, 18 Aug 2008 09:48:37 -0400 From: jim stasheff To: Categories list Subject: categories: Re: abutment 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: 20 Michael Barr wrote: > That seems to be the translation of the word, thanks to several. That > still leaves that mysterious E^{pq} ---> G^p(E^{p+q}) to correct, since I > do not believe that it is correct as stands. > > Michael > > Punctuation! the context ought to make it clear?? jim From rrosebru@mta.ca Mon Aug 18 19:26:23 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 18 Aug 2008 19:26:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVD9l-00044p-Lf for categories-list@mta.ca; Mon, 18 Aug 2008 19:24:57 -0300 Date: Mon, 18 Aug 2008 12:32:58 -0400 From: edubuc MIME-Version: 1.0 To: Categories list Subject: categories: Re: abutment 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: 21 I suggest to Michael Barr to send the whole sentence where it is found the word "aboutissement", and ask for a translation of the sentence. He will get a better english version that just a rendering into english of that word. May be the natural 21 century mathematical english version of the french sentence will not even use the word "abutment". Michael Barr wrote: > That seems to be the translation of the word, thanks to several. That > still leaves that mysterious E^{pq} ---> G^p(E^{p+q}) to correct, since I > do not believe that it is correct as stands. > > Michael > From rrosebru@mta.ca Tue Aug 19 13:35:33 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 19 Aug 2008 13:35:33 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVU9S-0003BQ-6f for categories-list@mta.ca; Tue, 19 Aug 2008 13:33:46 -0300 Message-ID: <48AA4AB3.8060700@dm.uba.ar> Date: Tue, 19 Aug 2008 01:23:15 -0300 From: "Eduardo J. Dubuc" MIME-Version: 1.0 To: Categories list Subject: categories: Re: abutment Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 22 Well Michael, I do not have at hand your original post. What I do remember of that post is that you ask for an english translation of the word "aboutissement", not for a translation a whole sentence containing that word. In my whole experience reading mathematics written in english, I do not remember to have seen the word "abutment". On the other hand, I am tired of reading spanish translations (from english or french or russian) of mathematical texts where single words are replaced by their spanish translations, and the result is completely alien to standard mathematical spanish. Does the word "abutment" belong to standard mathematical english ? Eduardo. Michael Barr > Maybe Eduardo should have read my whole post. > > Michael > > On Mon, 18 Aug 2008, edubuc wrote: > >> >> I suggest to Michael Barr to send the whole sentence where it is found >> the word "aboutissement", and ask for a translation of the sentence. >> He will get a better english version that just a rendering into english >> of that word. >> >> May be the natural 21 century mathematical english version of the french >> sentence will not even use the word "abutment". >> >> >> Michael Barr wrote: >>> That seems to be the translation of the word, thanks to several. That >>> still leaves that mysterious E^{pq} ---> G^p(E^{p+q}) to correct, >>> since I >>> do not believe that it is correct as stands. >>> >>> Michael >>> >> >> >> From rrosebru@mta.ca Tue Aug 19 13:35:33 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 19 Aug 2008 13:35:33 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVU8k-00037l-Ug for categories-list@mta.ca; Tue, 19 Aug 2008 13:33:02 -0300 Date: Mon, 18 Aug 2008 19:34:53 -0400 (EDT) From: Michael Barr To: Categories list Subject: categories: Re: abutment 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: 23 Maybe Eduardo should have read my whole post. Michael On Mon, 18 Aug 2008, edubuc wrote: > > I suggest to Michael Barr to send the whole sentence where it is found > the word "aboutissement", and ask for a translation of the sentence. > He will get a better english version that just a rendering into english > of that word. > > May be the natural 21 century mathematical english version of the french > sentence will not even use the word "abutment". > > > Michael Barr wrote: >> That seems to be the translation of the word, thanks to several. That >> still leaves that mysterious E^{pq} ---> G^p(E^{p+q}) to correct, since I >> do not believe that it is correct as stands. >> >> Michael >> > > > From rrosebru@mta.ca Tue Aug 19 13:58:00 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 19 Aug 2008 13:58:00 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVUWU-0005P5-2u for categories-list@mta.ca; Tue, 19 Aug 2008 13:57:34 -0300 Date: Tue, 19 Aug 2008 01:54:02 -0300 From: "Eduardo J. Dubuc" MIME-Version: 1.0 To: Categories list Subject: categories: Re: abutment Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 24 Well, I read my email backwards, so after my previous post I came into an email that it is certainly useful to many, including myself and Michael. This email tels me that "abutment" is the standard mathematical english version of the french word in the spectral sequence context as in The URL for this search is http://arxiv.org/find/grp_math/1/abs:+abutment/0/1/0/all/0/1 Michael Barr wrote: > Maybe Eduardo should have read my whole post. > > Michael > > On Mon, 18 Aug 2008, edubuc wrote: > >> >> I suggest to Michael Barr to send the whole sentence where it is found >> the word "aboutissement", and ask for a translation of the sentence. >> He will get a better english version that just a rendering into english >> of that word. >> >> May be the natural 21 century mathematical english version of the french >> sentence will not even use the word "abutment". >> >> >> Michael Barr wrote: >>> That seems to be the translation of the word, thanks to several. That >>> still leaves that mysterious E^{pq} ---> G^p(E^{p+q}) to correct, >>> since I >>> do not believe that it is correct as stands. >>> >>> Michael >>> >> >> >> From rrosebru@mta.ca Tue Aug 19 13:58:40 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 19 Aug 2008 13:58:40 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVUXP-0005Ut-Mq for categories-list@mta.ca; Tue, 19 Aug 2008 13:58:31 -0300 Date: Tue, 19 Aug 2008 08:28:20 +0200 Subject: categories: Re: abutment = aboutement? MIME-Version: 1.0 From: "mhebert" To: "categories" Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 25 Dear all, > That seems to be the translation [abutment] of the word, thanks to seve= ral. Just a remark about "abutment": it translates the French "aboutement", wi= th a rather different meaning than "aboutissement". The latter is closer= to the "ending" (of some process; with possibly a little shade of "fatal= ity" in it). The two words are related, and I don't know whether the mathematical idea= behind makes "abutment" good, or even better, but I just wanted to menti= on the difference. Michel Hebert Fromcat-dist@mta.ca To"Categories list" categories@mta.ca Cc DateSun, 17 Aug 2008 20:08:59 -0400 (EDT) Subjectcategories: abutment That > still leaves that mysterious E^{pq} ---> G^p(E^{p+q}) to correct, since= I > do not believe that it is correct as stands. > > Michael > > From rrosebru@mta.ca Tue Aug 19 13:59:01 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 19 Aug 2008 13:59:01 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVUXk-0005Wd-OJ for categories-list@mta.ca; Tue, 19 Aug 2008 13:58:52 -0300 Date: Tue, 19 Aug 2008 07:45:12 -0700 From: John Baez To: categories Subject: categories: spans in 2-categories 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: 26 Dear Categorists - Given a category C with pullbacks we can define a bicategory Span(C) where objects are objects of C, morphisms are spans - composed using pullback - and 2-morphisms are maps between spans. Have people tried to categorify this yet? Suppose we have a 2-category C with pseudo-pullbacks. Then we should be able to define a tricategory Span(C). Has someone done this? Or maybe people have gotten some partial results, e.g. in the case where C = Cat. I'd like to know about these! Best, jb From rrosebru@mta.ca Tue Aug 19 13:59:25 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 19 Aug 2008 13:59:25 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVUY5-0005Z2-7K for categories-list@mta.ca; Tue, 19 Aug 2008 13:59:13 -0300 Date: Tue, 19 Aug 2008 08:54:12 -0700 From: John Baez To: categories Subject: categories: biadjoint biequivalence 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: 27 Dear Categorists - I know someone who needs to construct a "biadjoint biequivalence" between bicategories, and is not relishing the task. If somebody was trying to construct an adjoint equivalence between categories, and was not relishing the task, I might suggest that they construct an equivalence, and then use the theorem that any equivalence can be improved to an adjoint equivalence. Has someone proved that any biequivalence between bicategories can be improved to a biadjoint biequivalence? Best, jb From rrosebru@mta.ca Tue Aug 19 22:02:23 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 19 Aug 2008 22:02:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVc4Z-0004fG-Vk for categories-list@mta.ca; Tue, 19 Aug 2008 22:01:16 -0300 Date: Tue, 19 Aug 2008 13:11:10 -0500 From: Michael Shulman To: categories Subject: categories: Re: biadjoint biequivalences and spans in 2-categories 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: 28 Hi John, Answers to both this and your previous question about biadjoint biequivalences are at least asserted in Street's "Fibrations in Bicategories". At the end of section 1, he defines a functor (=homomorphism) to have a left biadjoint if each object has a left bilifting, to be a biequivalence if it is biessentially surjective and locally fully faithful, and states that "clearly a biequivalence T has a left biadjoint S which is also a biequivalence". At the beginning of section 3 he defines a bicategory of spans from A to B in any bicategory, and given finite bilimits, essentially describes how to construct what one might call an "unbiased tricategory" of spans (of course, the definition of tricategory didn't exist at the time). He doesn't give any details of the proofs, but one could probably construct a detailed proof from these ideas without much more than tedium. I don't know whether anyone has written them out. Best, Mike On Tue, Aug 19, 2008 at 07:45:12AM -0700, John Baez wrote: > Dear Categorists - > > Given a category C with pullbacks we can define a bicategory Span(C) > where objects are objects of C, morphisms are spans - composed > using pullback - and 2-morphisms are maps between spans. > > Have people tried to categorify this yet? > > Suppose we have a 2-category C with pseudo-pullbacks. Then we should > be able to define a tricategory Span(C). Has someone done this? > > Or maybe people have gotten some partial results, e.g. in the case > where C = Cat. I'd like to know about these! > > Best, > jb > > > > From rrosebru@mta.ca Tue Aug 19 22:02:23 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 19 Aug 2008 22:02:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVc3y-0004dC-UL for categories-list@mta.ca; Tue, 19 Aug 2008 22:00:39 -0300 Date: Tue, 19 Aug 2008 11:06:19 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories Subject: categories: Re: abutment = aboutement? Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 29 I'm with Michel on this one: > Just a remark about "abutment": it translates the French "aboutement", > with a rather different meaning than "aboutissement". The latter is > closer to the "ending" (of some process; with possibly a little shade > of "fatality" in it). > > The two words are related, and I don't know whether the mathematical > idea behind makes "abutment" good, or even better, but I just wanted > to mention the difference. Not a single abutment in any of the following YouTube videos posted by their proud aboutisseurs. http://www.youtube.com/results?search_query=aboutissement&search_type= Evidently G needed a word with the sense of "limit" or "completion" that didn't overload terms that already had technical meanings in that context while itself having a technical ring to it, which "aboutissement" seems to do nicely in French. Something like "terminus" might serve this purpose in English. An abutment is an engineering construct for butting two things together, often in the context of bridges, whether over a river or between teeth, and seems quite unsuitable for this purpose. Vaughan From rrosebru@mta.ca Tue Aug 19 22:02:23 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 19 Aug 2008 22:02:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVc5V-0004j1-QY for categories-list@mta.ca; Tue, 19 Aug 2008 22:02:13 -0300 Date: Tue, 19 Aug 2008 17:31:17 -0400 From: jim stasheff MIME-Version: 1.0 To: Categories list Subject: categories: Re: abutment 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: 30 Eduardo J. Dubuc wrote: > Well Michael, I do not have at hand your original post. What I do > remember of > that post is that you ask for an english translation of the word > "aboutissement", not for a translation a whole sentence containing > that word. > > In my whole experience reading mathematics written in english, I do not > remember to have seen the word "abutment". > > On the other hand, I am tired of reading spanish translations (from > english or > french or russian) of mathematical texts where single words are > replaced by > their spanish translations, and the result is completely alien to > standard > mathematical spanish. > > Does the word "abutment" belong to standard mathematical english ? > > Eduardo. I thought the references I sent showed it is indeed standard mathematical english at least for those who speak the subdialect of spectral sequences even more common is the construction the _ spectral sequence abuts to.... jim From rrosebru@mta.ca Tue Aug 19 22:03:32 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 19 Aug 2008 22:03:32 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVc6e-0004oO-LH for categories-list@mta.ca; Tue, 19 Aug 2008 22:03:24 -0300 Date: Tue, 19 Aug 2008 19:23:03 -0500 From: Michael Shulman To: categories@mta.ca Subject: categories: symmetric monoidal traces 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: 31 Hi all, Can someone please point me to whatever categorical references exist regarding the canonical trace for dualizable objects in a symmetric monoidal category? I am particularly interested in (1) elementary expositions accessible to non-category-theorists and (2) any proofs of its uniqueness subject to various conditions. I know there are many references on traced monoidal categories, particularly with applications to computer science, but right now I am only interested in the symmetric monoidal trace. I also know that there are various reinventions/expositions of the notion in, for example, the topological literature (e.g. Dold-Puppe), but I would like an exposition not tied to any particular application. Finally, I know that the Joyal-Street-Verity paper "Traced Monoidal Categories" proves that the canonical symmetric (or, more precisely, balanced) monoidal trace is "universal" in that any traced monoidal category can be embedded in one equipped with the canonical trace, but as far as I can tell this need not determine the canonical trace uniquely. Thanks!! Mike From rrosebru@mta.ca Tue Aug 19 22:04:36 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 19 Aug 2008 22:04:36 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVc7f-0004sG-Id for categories-list@mta.ca; Tue, 19 Aug 2008 22:04:27 -0300 Date: Tue, 19 Aug 2008 13:11:34 -0400 From: "Ben Webster" Subject: categories: Re: abutment To: "Categories list" 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: 32 Yes, it does. Googling "abutment spectral sequence" makes this abundantly clear. Ben On Tue, Aug 19, 2008 at 12:23 AM, Eduardo J. Dubuc wrote: > Well Michael, I do not have at hand your original post. What I do remember > of > that post is that you ask for an english translation of the word > "aboutissement", not for a translation a whole sentence containing that > word. > > In my whole experience reading mathematics written in english, I do not > remember to have seen the word "abutment". > > On the other hand, I am tired of reading spanish translations (from english > or > french or russian) of mathematical texts where single words are replaced by > their spanish translations, and the result is completely alien to standard > mathematical spanish. > > Does the word "abutment" belong to standard mathematical english ? > > Eduardo. > > > > > Michael Barr >> >> Maybe Eduardo should have read my whole post. >> >> Michael >> >> On Mon, 18 Aug 2008, edubuc wrote: >> >>> >>> I suggest to Michael Barr to send the whole sentence where it is found >>> the word "aboutissement", and ask for a translation of the sentence. >>> He will get a better english version that just a rendering into english >>> of that word. >>> >>> May be the natural 21 century mathematical english version of the french >>> sentence will not even use the word "abutment". >>> >>> >>> Michael Barr wrote: >>>> >>>> That seems to be the translation of the word, thanks to several. That >>>> still leaves that mysterious E^{pq} ---> G^p(E^{p+q}) to correct, >>>> since I >>>> do not believe that it is correct as stands. >>>> >>>> Michael >>>> >>> >>> >>> > > > From rrosebru@mta.ca Wed Aug 20 08:22:35 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 20 Aug 2008 08:22:35 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVlkR-0001NR-Ry for categories-list@mta.ca; Wed, 20 Aug 2008 08:21:07 -0300 Date: Wed, 20 Aug 2008 02:12:44 -0300 (ART) Subject: categories: Re: abutment = aboutement? From: edubuc@dm.uba.ar To: "categories" 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: 33 I agree with Vaughan. Further, I have the feeling that "abutment" is not the appropriate way of rendering into mathematical english the meaning of the word "aboutissement" as it was used by Grothendieck. I repeat, we should analyse the whole french sentence to come up with a good translation. Is it not possible that somebody (not very versant in either french or english) had first the need to translate Grothendieck's "aboutissement", and unlike Michael Barr who asked advise, just came up with "abutment" (out of some dictionary). and then, other people (also not very good at either french or english) in the same area just keep copying him and each other? and generated the whole cascade coming out of google . . . who is to blame for the first use of "abutment" for Grothendieck's "aboutissement" in mathematical english ? ja !!! are we all going to follow ? I will be the first to use "abutment" if the word has a long tradition, and some prestigious mathematicians have used it. I finish with a question: Is it the case here ? Eduardo Dubuc > I'm with Michel on this one: > > > Just a remark about "abutment": it translates the French "aboutement", > > with a rather different meaning than "aboutissement". The latter is > > closer to the "ending" (of some process; with possibly a little shade > > of "fatality" in it). > > > > The two words are related, and I don't know whether the mathematical > > idea behind makes "abutment" good, or even better, but I just wanted > > to mention the difference. > > Not a single abutment in any of the following YouTube videos posted by > their proud aboutisseurs. > > http://www.youtube.com/results?search_query=aboutissement&search_type= > > Evidently G needed a word with the sense of "limit" or "completion" that > didn't overload terms that already had technical meanings in that > context while itself having a technical ring to it, which > "aboutissement" seems to do nicely in French. Something like "terminus" > might serve this purpose in English. > > An abutment is an engineering construct for butting two things together, > often in the context of bridges, whether over a river or between teeth, > and seems quite unsuitable for this purpose. > > Vaughan > > From rrosebru@mta.ca Wed Aug 20 08:22:35 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 20 Aug 2008 08:22:35 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVll6-0001Px-Av for categories-list@mta.ca; Wed, 20 Aug 2008 08:21:48 -0300 Date: Wed, 20 Aug 2008 07:56:44 +0100 (BST) From: Richard Garner To: categories Subject: categories: Re: biadjoint biequivalences and spans in 2-categories 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: 34 > At the beginning of section 3 he defines a bicategory of spans from A to > B in any bicategory, and given finite bilimits, essentially describes > how to construct what one might call an "unbiased tricategory" of spans > (of course, the definition of tricategory didn't exist at the time). > > He doesn't give any details of the proofs, but one could probably > construct a detailed proof from these ideas without much more than > tedium. I don't know whether anyone has written them out. One amusing thing about the tricategory of spans in a 2-category, with composition by iso-comma object, is that it is barely a tricategory. Binary composition is associative up to iso; it is only the unitality which is really up to equivalence. Something similar happens if you define a tricategory of biprofunctors. Richard From rrosebru@mta.ca Wed Aug 20 08:22:36 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 20 Aug 2008 08:22:36 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVllj-0001SO-Md for categories-list@mta.ca; Wed, 20 Aug 2008 08:22:27 -0300 From: To: carlos.martin@urv.cat Date: Wed, 20 Aug 2008 10:58:55 +0200 MIME-Version: 1.0 Content-Language: ca Subject: categories: LATA 2009: 2nd call for papers Content-Type: text/plain; charset=windows-1252 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: 35 Apologies for multiple posting! Please=2C forward the announcement to whoever may be interested in it=2E= Thanks=2E ********************************************************************* Second Call for Papers 3rd INTERNATIONAL CONFERENCE ON LANGUAGE AND AUTOMATA THEORY AND APPLICA= TIONS (LATA 2009) Tarragona=2C Spain=2C April 2-8=2C 2009 http=3A//grammars=2Egrlmc=2Ecom/LATA2009/ ********************************************************************* AIMS=3A LATA is a yearly conference in theoretical computer science and its appl= ications=2E As linked to the International PhD School in Formal Language= s and Applications that was developed at the host institute in the perio= d 2002-2006=2C LATA 2009 will reserve significant room for young scholar= s at the beginning of their career=2E It will aim at attracting contribu= tions from both classical theory fields and application areas (bioinform= atics=2C systems biology=2C language technology=2C artificial intelligen= ce=2C etc=2E)=2E SCOPE=3A Topics of either theoretical or applied interest include=2C but are not = limited to=3A - algebraic language theory - algorithms on automata and words - automata and logic - automata for system analysis and programme verification - automata=2C concurrency and Petri nets - biomolecular nanotechnology - cellular automata - circuits and networks - combinatorics on words - computability - computational=2C descriptional=2C communication and parameterized comp= lexity - data and image compression - decidability questions on words and languages - digital libraries - DNA and other models of bio-inspired computing - document engineering - extended automata - foundations of finite state technology - fuzzy and rough languages - grammars (Chomsky hierarchy=2C contextual=2C multidimensional=2C unifi= cation=2C categorial=2C etc=2E) - grammars and automata architectures - grammatical inference and algorithmic learning - graphs and graph transformation - language varieties and semigroups - language-based cryptography - language-theoretic foundations of natural language processing=2C artif= icial intelligence and artificial life - mathematical evolutionary genomics - parsing - patterns and codes - power series - quantum=2C chemical and optical computing - regulated rewriting - string and combinatorial issues in computational biology and bioinform= atics - symbolic dynamics - symbolic neural networks - term rewriting - text algorithms - text retrieval=2C pattern matching and pattern recognition - transducers - trees=2C tree languages and tree machines - weighted machines STRUCTURE=3A LATA 2009 will consist of=3A - 3 invited talks - 2 invited tutorials - refereed contributions - open sessions for discussion in specific subfields or on professional = issues (if requested by the participants) Invited speakers will be=3A Bruno Courcelle (Bordeaux)=3A Graph Structure and Monadic Second-order L= ogic (tutorial) Markus Holzer (Muenchen)=3A Nondeterministic Finite Automata=3A Recent D= evelopments (tutorial) Sanjay Jain (Singapore)=3A Role of Hypothesis Spaces in Inductive Infere= nce Kai Salomaa (Kingston=2C Canada)=3A State Complexity of Nested Word Auto= mata Thomas Zeugmann (Sapporo)=3A Recent Developments in Algorithmic Teaching= PROGRAMME COMMITTEE=3A Parosh Abdulla (Uppsala) Stefania Bandini (Milano) Stephen Bloom (Hoboken) John Brzozowski (Waterloo) Maxime Crochemore (London) Juergen Dassow (Magdeburg) Michael Domaratzki (Winnipeg) Henning Fernau (Trier) Rusins Freivalds (Riga) Vesa Halava (Turku) Juraj Hromkovic (Zurich) Lucian Ilie (London=2C Canada) Kazuo Iwama (Kyoto) Aravind Joshi (Philadelphia) Juhani Karhumaki (Turku) Jarkko Kari (Turku) Claude Kirchner (Bordeaux) Maciej Koutny (Newcastle) Hans-Joerg Kreowski (Bremen) Kamala Krithivasan (Chennai) Martin Kutrib (Giessen) Andrzej Lingas (Lund) Aldo de Luca (Napoli) Rupak Majumdar (Los Angeles) Carlos Martin-Vide (Tarragona =26 Brussels=2C chair) Joachim Niehren (Lille) Antonio Restivo (Palermo) Joerg Rothe (Duesseldorf) Wojciech Rytter (Warsaw) Philippe Schnoebelen (Cachan) Thomas Schwentick (Dortmund) Helmut Seidl (Muenchen) Alan Selman (Buffalo) Jeffrey Shallit (Waterloo) Ludwig Staiger (Halle) Frank Stephan (Singapore) ORGANIZING COMMITTEE=3A Madalina Barbaiani Gemma Bel-Enguix Cristina Bibire Adrian-Horia Dediu Szilard-Zsolt Fazekas Armand-Mihai Ionescu M=2E Dolores Jimenez-Lopez Alexander Krassovitskiy Guangwu Liu Carlos Martin-Vide (chair) Zoltan-Pal Mecsei Robert Mercas Catalin-Ionut Tirnauca Bianca Truthe Sherzod Turaev Florentina-Lilica Voicu SUBMISSIONS=3A Authors are invited to submit papers presenting original and unpublished= research=2E Papers should not exceed 12 single-spaced pages and should = be formatted according to the standard format for Springer Verlag=27s LN= CS series (see http=3A//www=2Espringer=2Ecom/computer/lncs/lncs+authors=3F= SGWID=3D0-40209-0-0-0)=2E Submissions have to be uploaded at=3A http=3A//www=2Eeasychair=2Eorg/conferences/=3Fconf=3Dlata2009 PUBLICATIONS=3A A volume of proceedings published by Springer in the LNCS series will be= available by the time of the conference=2E Two special issues of the journals Information and Computation (Elsevier= =2C 2007 impact factor=3A 0=2E983) and Journal of Logic and Computation = (Oxford University Press=2C 2007 impact factor=3A 0=2E821) containing ex= tended versions of selected papers will be published after the conferenc= e=2E REGISTRATION=3A The period for registration will be open since September 1=2C 2008 to Ap= ril 2=2C 2009=2E The registration form can be found at the website of th= e conference=3A http=3A//grammars=2Egrlmc=2Ecom/LATA2009/ Early registration fees=3A 450 euros Early registration fees (PhD students)=3A 225 euros Registration fees=3A 540 euros Registration fees (PhD students)=3A 270 euros At least one author per paper should register=2E Papers that do not have= a registered author by December 31=2C 2008 will be excluded from the pr= oceedings=2E Fees comprise access to all sessions=2C one copy of the proceedings volu= me=2C and coffee breaks=2E For the participation in the full-day excursi= on and conference lunch on Sunday April 5=2C the amount of 70 euros is t= o be added to the fees above=3A accompanying persons are welcome at the = same rate=2E PAYMENT=3A Early registration fees must be paid by bank transfer before December 31= =2C 2008 to the conference account at Open Bank (Plaza Manuel Gomez More= no 2=2C 28020 Madrid=2C Spain)=3A IBAN=3A ES1300730100510403506598 - Swi= ft code=3A OPENESMMXXX (account holder=3A LATA 2009 =96 Carlos Martin-Vi= de)=2E (Non-early) registration fees can be paid either by bank transfer to the= same account or in cash on site=2E Besides paying the registration fees=2C it is required to fill in the re= gistration form at the website of the conference=2E A receipt for the pa= yment will be provided on site=2E FUNDING=3A Up to 20 grants covering partial-board accommodation will be available f= or nonlocal PhD students=2E To apply=2C candidates must e-mail their CV = together with a copy of the document proving their present status as a P= hD student=2E IMPORTANT DATES=3A Paper submission=3A October 22=2C 2008 Notification of paper acceptance or rejection=3A December 10=2C 2008 Application for funding (PhD students)=3A December 15=2C 2008 Notification of funding acceptance or rejection=3A December 19=2C 2008 Final version of the paper for the proceedings=3A December 24=2C 2008 Early registration=3A December 31=2C 2008 Starting of the conference=3A April 2=2C 2009 Submission to the journal special issues=3A June 22=2C 2009 FURTHER INFORMATION=3A carlos=2Emartin=40urv=2Ecat ADDRESS=3A LATA 2009 Research Group on Mathematical Linguistics Rovira i Virgili University Plaza Imperial Tarraco=2C 1 43005 Tarragona=2C Spain Phone=3A +34-977-559543 Fax=3A +34-977-559597 From rrosebru@mta.ca Wed Aug 20 08:23:13 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 20 Aug 2008 08:23:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVlmL-0001VQ-N2 for categories-list@mta.ca; Wed, 20 Aug 2008 08:23:05 -0300 Date: Wed, 20 Aug 2008 04:04:12 -0500 (CDT) From: Tom Fiore To: categories@mta.ca Subject: categories: Re: biadjoint biequivalences and spans in 2-categories 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: 36 Hello, Details of a part of John and Mike's post can be found in one of my publications. Theorem. 9.17 Let X and A be strict 2-categories, and G:A -> X a pseudo functor. There exists a left biadjoint for G if and only if for every object x of X there exists an object r of A and a biuniversal arrow x -> Gr from x to G. The proof is as one could expect. http://arxiv.org/abs/math.CT/0408298 Fiore, Thomas M. Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory. Mem. Amer. Math. Soc. 182 (2006), no. 860, x+171 pp. Best greetings, Tom On Tue, 19 Aug 2008, Michael Shulman wrote: > Hi John, > > Answers to both this and your previous question about biadjoint > biequivalences are at least asserted in Street's "Fibrations in > Bicategories". > > At the end of section 1, he defines a functor (=homomorphism) to have a > left biadjoint if each object has a left bilifting, to be a biequivalence > if it is biessentially surjective and locally fully faithful, and states > that "clearly a biequivalence T has a left biadjoint S which is also a > biequivalence". > > At the beginning of section 3 he defines a bicategory of spans from A to > B in any bicategory, and given finite bilimits, essentially describes > how to construct what one might call an "unbiased tricategory" of spans > (of course, the definition of tricategory didn't exist at the time). > > He doesn't give any details of the proofs, but one could probably > construct a detailed proof from these ideas without much more than > tedium. I don't know whether anyone has written them out. > > Best, > Mike > > On Tue, Aug 19, 2008 at 07:45:12AM -0700, John Baez wrote: >> Dear Categorists - >> >> Given a category C with pullbacks we can define a bicategory Span(C) >> where objects are objects of C, morphisms are spans - composed >> using pullback - and 2-morphisms are maps between spans. >> >> Have people tried to categorify this yet? >> >> Suppose we have a 2-category C with pseudo-pullbacks. Then we should >> be able to define a tricategory Span(C). Has someone done this? >> >> Or maybe people have gotten some partial results, e.g. in the case >> where C = Cat. I'd like to know about these! >> >> Best, >> jb >> >> >> >> > > From rrosebru@mta.ca Thu Aug 21 09:36:33 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 21 Aug 2008 09:36:33 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KW9MO-0006nD-M0 for categories-list@mta.ca; Thu, 21 Aug 2008 09:33:52 -0300 Date: Wed, 20 Aug 2008 21:39:59 +0930 From: David Roberts To: categories@mta.ca Subject: categories: Re: biadjoint biequivalences 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: 37 Hi all, Tom Fiore wrote: > Theorem. 9.17 > Let X and A be strict 2-categories, and G:A -> X a pseudo functor. Ther= e > exists a left biadjoint for G if and only if for every object x of X th= ere > exists an object r of A and a biuniversal arrow x -> Gr from x to G. Of course this begs the obvious question, how hard is this to generalise = to bicategories? I'm surprised no-one has mentioned Gurksi's thesis, which I just came acr= oss. Appendix A has details of adjunctions in bicategories, and biadjunctions = in tricategories, citing Verity's thesis in the case of Gray-categories. Best, David From rrosebru@mta.ca Thu Aug 21 09:36:33 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 21 Aug 2008 09:36:33 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KW9O3-0006vO-Ha for categories-list@mta.ca; Thu, 21 Aug 2008 09:35:35 -0300 Date: Wed, 20 Aug 2008 10:45:36 -0400 From: jim stasheff MIME-Version: 1.0 To: categories Subject: categories: Re: abutment = aboutement? 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: 38 > An abutment is an engineering construct for butting two things together, > often in the context of bridges, whether over a river or between teeth, > and seems quite unsuitable for this purpose. > > Vaughan I admit I don't recall abutment but abutting has a long history in spec sequence jargon the idea I think was that the spectral sequence runs to/ buts up against the E\infty term or is it to the object of which E\infty is the associated graded? sorry it's been so long... jim > > > > From rrosebru@mta.ca Thu Aug 21 09:36:33 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 21 Aug 2008 09:36:33 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KW9N9-0006rQ-HK for categories-list@mta.ca; Thu, 21 Aug 2008 09:34:39 -0300 Date: Wed, 20 Aug 2008 09:13:26 -0400 (EDT) From: Michael Barr To: categories Subject: categories: Re: abutment = aboutement? 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: 39 I am NOT about to change a word that has apparently existed for over 50 years just because it is not particularly meaningful. Does "ring", which originally referred to Z/nZ, still have any connection with the notion? How about "field" or as the French and Germans call them "body" have any connection with fields. The worst thing is to create a fork in the language. I have my own idea on what a better word would be but I'll be damned if I state it in a public forum. Anyway, I gather that Grothendieck's definition is hardly used today. It is interesting that the less important question (and ultimately unimportant) question has flooded the list, while the serious one (what was Grothendieck's supposed to be) has been ignored. I have no suggestion, save my own, for what definition was actually intended. Michael On Wed, 20 Aug 2008, edubuc@dm.uba.ar wrote: > > > I agree with Vaughan. > > Further, I have the feeling that "abutment" is not the appropriate way of > rendering into mathematical english the meaning of the word > "aboutissement" as it was used by Grothendieck. > > I repeat, we should analyse the whole french sentence to come up with a > good translation. > > Is it not possible that somebody (not very versant in either french or > english) had first the need to translate Grothendieck's "aboutissement", > and unlike Michael Barr who asked advise, just came up with "abutment" > (out of some dictionary). > > and then, other people (also not very good at either french or english) > in the same area just keep copying him and each other? > > and generated the whole cascade coming out of google . . . > > who is to blame for the first use of "abutment" for Grothendieck's > "aboutissement" in mathematical english ? ja !!! > > are we all going to follow ? > > I will be the first to use "abutment" if the word has a long tradition, > and some prestigious mathematicians have used it. > > I finish with a question: Is it the case here ? > > Eduardo Dubuc > >> I'm with Michel on this one: >> >> > Just a remark about "abutment": it translates the French "aboutement", >> > with a rather different meaning than "aboutissement". The latter is >> > closer to the "ending" (of some process; with possibly a little shade >> > of "fatality" in it). >> > >> > The two words are related, and I don't know whether the mathematical >> > idea behind makes "abutment" good, or even better, but I just wanted >> > to mention the difference. >> >> Not a single abutment in any of the following YouTube videos posted by >> their proud aboutisseurs. >> >> http://www.youtube.com/results?search_query=aboutissement&search_type= >> >> Evidently G needed a word with the sense of "limit" or "completion" that >> didn't overload terms that already had technical meanings in that >> context while itself having a technical ring to it, which >> "aboutissement" seems to do nicely in French. Something like "terminus" >> might serve this purpose in English. >> >> An abutment is an engineering construct for butting two things together, >> often in the context of bridges, whether over a river or between teeth, >> and seems quite unsuitable for this purpose. >> >> Vaughan >> >> > > > > From rrosebru@mta.ca Thu Aug 21 09:36:40 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 21 Aug 2008 09:36:40 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KW9Oy-00070n-7j for categories-list@mta.ca; Thu, 21 Aug 2008 09:36:32 -0300 Date: Wed, 20 Aug 2008 11:33:10 -0400 From: jim stasheff MIME-Version: 1.0 To: categories Subject: categories: Re: abutment = aboutement? Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 40 Vaughan Pratt wrote: > I'm with Michel on this one: > > > Just a remark about "abutment": it translates the French "aboutement", > > with a rather different meaning than "aboutissement". The latter is > > closer to the "ending" (of some process; with possibly a little shade > > of "fatality" in it). > > > > The two words are related, and I don't know whether the mathematical > > idea behind makes "abutment" good, or even better, but I just wanted > > to mention the difference. > > Not a single abutment in any of the following YouTube videos posted by > their proud aboutisseurs. > > http://www.youtube.com/results?search_query=aboutissement&search_type= > > Evidently G needed a word with the sense of "limit" or "completion" that > didn't overload terms that already had technical meanings in that > context while itself having a technical ring to it, which > "aboutissement" seems to do nicely in French. Something like "terminus" > might serve this purpose in English. > > An abutment is an engineering construct for butting two things together, > often in the context of bridges, whether over a river or between teeth, > and seems quite unsuitable for this purpose. > > Vaughan > > Mathematicians have a gift for language - not to worry about translation consider translating `field' into French Russian german ... From rrosebru@mta.ca Thu Aug 21 09:46:42 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 21 Aug 2008 09:46:42 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KW9YU-00004s-4l for categories-list@mta.ca; Thu, 21 Aug 2008 09:46:22 -0300 Subject: categories: Re: symmetric monoidal traces Date: Wed, 20 Aug 2008 13:22:57 -0300 (ADT) To: categories@mta.ca MIME-Version: 1.0 From: selinger@mathstat.dal.ca (Peter Selinger) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 41 [ Reminder from moderator: Attachments are not suitable for transmission and the one mentioned below has been deleted (happily Peter provides a url). On a similar note, recall that html messages are not suitable either, and will not be posted. Please send text only. Thanks. ] Dear Mike, I don't know whether the proof of this result is spelled out in the literature (uniqueness of the trace for dualizable objects in a symmetric monoidal category). However, I have seen the result itself mentioned, and it follows straightforwardly from Joyal-Street-Verity's INT construction. As you have already said, every traced symmetric monoidal category C can be embedded in a compact closed category INT(C), in such a way that the trace of C is mapped to the canonical trace of INT(C). Further, every strong monoidal functor preserves dual objects, so if some object X of C has a "canonical" trace coming from a dual object, then this also gets mapped to the canonical trace of INT(C). Finally, since the functor is faithful, and maps the "given" and the "canonical" trace to the same thing, it follows that the two traces already coincide in C. The same argument works for balanced monoidal categories. One can easily turn this argument into an elementary algebraic proof. For an object X equipped with two traces Tr and Tr', consider the following "interchange property" for f: A*X*X -> B*X*X: Tr'_X(Tr_X(f o (A*c))) = Tr_X(Tr'_X((B*c) o f)) See Figure (a) in the attached file for an illustration of this property. It is akin to "symmetry sliding", except that it uses two different traces. Figure (b) proves that if two traces satisfy the interchange property, then they coincide. Finally, if one of the traces is the canonical one obtained from a dual of X, then the interchange property holds by standard diagrammatic reasoning (see Figure (c)). In particular, if a dual exists, then the trace on X is unique. The proof, as shown in the attachment, is only correct in the symmetric case. It also works in the balanced case, provided that one inserts a twist map in the correct places. -- Peter (Attachment also available as http://www.mathstat.dal.ca/~selinger/downloads/traces.gif) Michael Shulman wrote: > > Hi all, > > Can someone please point me to whatever categorical references exist > regarding the canonical trace for dualizable objects in a symmetric > monoidal category? I am particularly interested in (1) elementary > expositions accessible to non-category-theorists and (2) any proofs of > its uniqueness subject to various conditions. > > I know there are many references on traced monoidal categories, > particularly with applications to computer science, but right now I am > only interested in the symmetric monoidal trace. I also know that > there are various reinventions/expositions of the notion in, for > example, the topological literature (e.g. Dold-Puppe), but I would > like an exposition not tied to any particular application. Finally, I > know that the Joyal-Street-Verity paper "Traced Monoidal Categories" > proves that the canonical symmetric (or, more precisely, balanced) > monoidal trace is "universal" in that any traced monoidal category can > be embedded in one equipped with the canonical trace, but as far as I > can tell this need not determine the canonical trace uniquely. > > Thanks!! > Mike From rrosebru@mta.ca Thu Aug 21 09:48:27 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 21 Aug 2008 09:48:27 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KW9aN-0000HV-Dw for categories-list@mta.ca; Thu, 21 Aug 2008 09:48:19 -0300 Date: Thu, 21 Aug 2008 03:15:13 -0300 From: "Eduardo J. Dubuc" MIME-Version: 1.0 To: Categories list Subject: categories: Re: abutment = aboutement? 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: 42 Fred Linton has in this mail enlighten us all about the meaning in french of the words "aboutissement" and "aboutement". Quite different meanings. (*) It seems that "aboutir" means more or less "to arrive" or "to finish" "come to the end" etc, while "abuter" means 'to join end to end', thus ""abuter" a path" would mean "to make it into a loop" while ""aboutir" a path" would mean "to arrive to the end point" if "abutment" translates "aboutement", it certaily does not translate "aboutissement" Having in mind (*) above, somebody knowledgeable in both mathematics (in particular spectral sequences) and english should be able to come up with a correct english version of what Grothendieck meant by "aboutissement", which was certainly not "abutment". Fred E.J. Linton wrote: > Greetings. > > Way back on Monday, in an email direct to Mike, I had asked, > >> Think 'abut' for aboutir or 'abutment' for aboutissement >> are unusable false cognates? > > The more I read the comments here, and the more I consult > dictionaries and phrase books, the more I come to think > the answer, alas, is YES. > > Without, for the moment, examining the roles of "aboutir" > and "aboutissement" in the setting of spectral sequences, > let me expound for a bit on plain French philology. > > French "bout" is a masculine noun whose meaning tends to be > along the lines of 'end', 'tip', 'extremity', as in the > idiomatic expressions: > > "aux bouts du monde" = 'to the ends of the earth' ; > "d'un bout à l'autre" = 'from beginning to end' > (literally: 'from one end to the other') ; > "sur le bout des doigts" = 'at [one's] finger tips' ; > "au bout d'une heure" = 'after (lit.: at the end of) an hour' ; > "le bout de la langue" = 'the tip (extremity) of the tongue' ; > "au bout de la rue" = 'at the end of the street' . > > Not to be confused with French "but" = 'end' in the rather different > sense of 'goal', 'aim', 'target', 'purpose', etc. > > French "abouter" is a verb, derived from "bout", whose meaning is > 'to join (or to place) end to end'; its past participle "abouté" > thus serves as the adjective 'placed (or joined) end to end', > whence the carpenterial nouns "about", for 'end' or 'butt-end' > and "aboutement", 'butt-junction' or 'abutment'. > > The French verb "aboutir" develops 'end' rather differently: its > meanings are rather 'to come to an end at (or with)', 'to join', > 'to meet', 'to border upon', 'to end in', 'to tend to', as with: > > "N'aboutir à rien" = 'to come to nothing' ; > "Ce champ aboutit à un marais" = 'this field borders upon a swamp' ; > the long, winding road that may quite possibly "aboutit" in a cul-de-sac ; > "N'aboutir à rien" = 'to come to nothing' ; > and even "Faire aboutir un abscè" = 'to bring an abscess to a head' > (literally: 'to cause an abscess to come to an end') . > > The French noun "aboutissement", being derived from "aboutir" > rather than from "abouter", thus differs from "aboutement" in > signifying rather the end (or border or new state or condition) > at or to which something may "aboutit"; or, the act of achieving > that end, border, state, or condition. In particular, for a context > where "aboutir" has the meaning 'to tend to', the sense of > "aboutissement" may well be 'that which is tended to', which > is very nearly 'limit'. This nearly suggests that "aboutir" might > even, at times, be capable of expressing the sense 'to converge to'. > > That said, how does this all fit with spectral sequences? > > Certainly there is well-established usage involving the terms > of a spectral sequence converging to something-or-other. > Only in the full context of the French text around "aboutir" > or "aboutissement", though, would I be able to hazard any guess > whether 'convergence' or 'limit' would be the right counterparts. > Quite possibly they are *not*. > > Still, I'm hesitant to withdraw my warning that 'abut' and > 'abutment' are probably false cognates, no matter that several > mathematical authors have chosen to use them to render these terms. > > Is there any hope, perhaps, of getting input from some francophone > spectral sequence experts -- best of all, from AG himself? > > Thanks to Eduardo D and Vaughan P and Michel H for their misgivings, > which encouraged me to compose the above, despite the assurances > of Jim S that the 'abut*' usage is by now well entrenched. > > Cheers, > > -- Fred > > ------ Original Message ------ > Received: Wed, 20 Aug 2008 07:27:05 AM EDT > From: edubuc@dm.uba.ar > To: "categories" > Subject: Re: categories: Re: Re: abutment = aboutement? > >> >> I agree with Vaughan. >> >> Further, I have the feeling that "abutment" is not the appropriate way of >> rendering into mathematical english the meaning of the word >> "aboutissement" as it was used by Grothendieck. >> >> I repeat, we should analyse the whole french sentence to come up with a >> good translation. >> >> Is it not possible that somebody (not very versant in either french or >> english) had first the need to translate Grothendieck's "aboutissement", >> and unlike Michael Barr who asked advise, just came up with "abutment" >> (out of some dictionary). >> >> and then, other people (also not very good at either french or english) >> in the same area just keep copying him and each other? >> >> and generated the whole cascade coming out of google . . . >> >> who is to blame for the first use of "abutment" for Grothendieck's >> "aboutissement" in mathematical english ? ja !!! >> >> are we all going to follow ? >> >> I will be the first to use "abutment" if the word has a long tradition, >> and some prestigious mathematicians have used it. >> >> I finish with a question: Is it the case here ? >> >> Eduardo Dubuc >> >>> I'm with Michel on this one: >>> >>> > Just a remark about "abutment": it translates the French "aboutement", >>> > with a rather different meaning than "aboutissement". The latter is >>> > closer to the "ending" (of some process; with possibly a little shade >>> > of "fatality" in it). >>> > >>> > The two words are related, and I don't know whether the mathematical >>> > idea behind makes "abutment" good, or even better, but I just wanted >>> > to mention the difference. >>> >>> Not a single abutment in any of the following YouTube videos posted by >>> their proud aboutisseurs. >>> >>> http://www.youtube.com/results?search_query=aboutissement&search_type= >>> >>> Evidently G needed a word with the sense of "limit" or "completion" that >>> didn't overload terms that already had technical meanings in that >>> context while itself having a technical ring to it, which >>> "aboutissement" seems to do nicely in French. Something like "terminus" >>> might serve this purpose in English. >>> >>> An abutment is an engineering construct for butting two things together, >>> often in the context of bridges, whether over a river or between teeth, >>> and seems quite unsuitable for this purpose. >>> >>> Vaughan >>> >>> >> >> >> >> > > From rrosebru@mta.ca Thu Aug 21 19:22:19 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 21 Aug 2008 19:22:19 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KWIW9-00008k-Ja for categories-list@mta.ca; Thu, 21 Aug 2008 19:20:33 -0300 Subject: categories: Re: symmetric monoidal traces To: categories@mta.ca (Categories List) Date: Thu, 21 Aug 2008 10:39:08 -0300 (ADT) MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: <20080821133908.A644E5C2A8@chase.mathstat.dal.ca> From: selinger@mathstat.dal.ca (Peter Selinger) Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 43 As Phil Scott immediately pointed out to me, the proof of this folklore result appears in a 2000 slide by Masahito Hasegawa entitled "A short proof of the uniqueness of trace on tortile categories", see http://www.kurims.kyoto-u.ac.jp/~hassei/papers/canonicaltrace.gif and also on p.23 in a paper by the same author that is to appear in MSCS, entitled "On traced monoidal closed categories": http://www.kurims.kyoto-u.ac.jp/~hassei/papers/tmcc-revised16may08.pdf -- Peter Peter Selinger wrote: > > [ Reminder from moderator: Attachments are not suitable for transmission > and the one mentioned below has been deleted (happily Peter provides a > url). On a similar note, recall that html messages are not suitable > either, and will not be posted. Please send text only. Thanks. ] > > Dear Mike, > > I don't know whether the proof of this result is spelled out in the > literature (uniqueness of the trace for dualizable objects in a > symmetric monoidal category). However, I have seen the result itself > mentioned, and it follows straightforwardly from Joyal-Street-Verity's > INT construction. > > As you have already said, every traced symmetric monoidal category C > can be embedded in a compact closed category INT(C), in such a way > that the trace of C is mapped to the canonical trace of > INT(C). Further, every strong monoidal functor preserves dual objects, > so if some object X of C has a "canonical" trace coming from a dual > object, then this also gets mapped to the canonical trace of INT(C). > Finally, since the functor is faithful, and maps the "given" and the > "canonical" trace to the same thing, it follows that the two traces > already coincide in C. The same argument works for balanced monoidal > categories. > > One can easily turn this argument into an elementary algebraic > proof. For an object X equipped with two traces Tr and Tr', consider > the following "interchange property" for f: A*X*X -> B*X*X: > > Tr'_X(Tr_X(f o (A*c))) = Tr_X(Tr'_X((B*c) o f)) > > See Figure (a) in the attached file for an illustration of this > property. It is akin to "symmetry sliding", except that it uses two > different traces. > > Figure (b) proves that if two traces satisfy the interchange property, > then they coincide. > > Finally, if one of the traces is the canonical one obtained from a > dual of X, then the interchange property holds by standard > diagrammatic reasoning (see Figure (c)). In particular, if a dual > exists, then the trace on X is unique. > > The proof, as shown in the attachment, is only correct in the > symmetric case. It also works in the balanced case, provided that one > inserts a twist map in the correct places. > > -- Peter > > (Attachment also available as > http://www.mathstat.dal.ca/~selinger/downloads/traces.gif) > > Michael Shulman wrote: > > > > Hi all, > > > > Can someone please point me to whatever categorical references exist > > regarding the canonical trace for dualizable objects in a symmetric > > monoidal category? I am particularly interested in (1) elementary > > expositions accessible to non-category-theorists and (2) any proofs of > > its uniqueness subject to various conditions. > > > > I know there are many references on traced monoidal categories, > > particularly with applications to computer science, but right now I am > > only interested in the symmetric monoidal trace. I also know that > > there are various reinventions/expositions of the notion in, for > > example, the topological literature (e.g. Dold-Puppe), but I would > > like an exposition not tied to any particular application. Finally, I > > know that the Joyal-Street-Verity paper "Traced Monoidal Categories" > > proves that the canonical symmetric (or, more precisely, balanced) > > monoidal trace is "universal" in that any traced monoidal category can > > be embedded in one equipped with the canonical trace, but as far as I > > can tell this need not determine the canonical trace uniquely. > > > > Thanks!! > > Mike > > From rrosebru@mta.ca Fri Aug 22 09:34:10 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 22 Aug 2008 09:34:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KWVpp-0005oL-NA for categories-list@mta.ca; Fri, 22 Aug 2008 09:33:45 -0300 From: Nimish Shah To: categories Subject: categories: Re: abutment = aboutement? Date: Thu, 21 Aug 2008 15:30:52 +0100 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: 44 On Wednesday 20 August 2008 2:13 pm, Michael Barr wrote: > I am NOT about to change a word that has apparently existed for > over 50 years just because it is not particularly meaningful. I am going to risk putting my two cents here and perhaps being rejected by the moderator. One of the reasons why a word may need to be changed even if it has existed for 50 years is to avoid confusion. For example in the book "Categories for Software Engineering" the author talks about the "social life" of a set being the other sets it talks to. For a long while this puzzled me, until it dawned on me that the idea that the author was using was that the origins of modern Object-Orientated Programming (ie C++, Java) started with SmallTalk. In SmallTalk, objects communicated with one another by sending messages; and so making an analogy with familiar concepts that programmers use "sets have a social life" because SW objects "talk" to each other by sending messages. Stated another way it is difficult to see how a (Mathematical) object is the same as a (Software) object. The latter gets created and destroyed as the object comes in and out of scope. Hence what does one do? When stated generically, does the word object mean a mathematical one (around 60 years old) or a software one (more widely used)? Nim. From rrosebru@mta.ca Fri Aug 22 09:34:10 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 22 Aug 2008 09:34:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KWVoZ-0005jj-St for categories-list@mta.ca; Fri, 22 Aug 2008 09:32:27 -0300 Date: Thu, 21 Aug 2008 15:07:55 +0100 From: Tim Porter To: Categories list Subject: categories: Re: abutment = aboutement? 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: 45 Following the webpage:=0A= http://fr.wiktionary.org/wiki/aboutissement=0A= =0A= perhaps the word outcome suggests itself.=0A= =0A= Tim=0A= =0A= =0A= =0A= =0A= =0A= Quoting "Eduardo J. Dubuc" :=0A= =0A= > Fred Linton has in this mail enlighten us all about the meaning in french= of=0A= > the words "aboutissement" and "aboutement". Quite different meanings.=0A= >=0A= > (*) It seems that "aboutir" means more or less "to arrive" or "to finish"= =0A= > "come to the end" etc,=0A= >=0A= ... From rrosebru@mta.ca Fri Aug 22 09:35:14 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 22 Aug 2008 09:35:14 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KWVqc-0005s8-TO for categories-list@mta.ca; Fri, 22 Aug 2008 09:34:34 -0300 Date: Thu, 21 Aug 2008 12:18:09 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: Categories list Subject: categories: Re: abutment = aboutement? 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: 46 Meanwhile I count eight occurrences of "abut" and "abutment" in the (36 kilobyte!) main Wikipedia article on spectral sequences (there are a dozen separate much shorter articles on particular spectral sequences, along with a 15 kB article on derived categories). On the other hand the algebra and geometry articles of the 1987 Britannica Macropaedia both prefer the term "limit" for what a spectral sequence converges to, in respectively Peter Hilton's contribution "Other aspects of homological algebra" to the algebra article, and the geometry article's section on algebraic topology. Since Wikipedia seems to be trumping Britannica these days, and no one here has objected to established usage in mathematics trumping linguistic suitability, the precise distance of "abutment" from the optimal English cognate for "aboutissement" would appear to be academic, an epithet reflecting the outside world's perception that raising moot points is in our job description. Vaughan >> Thanks to Eduardo D and Vaughan P and Michel H for their misgivings, >> which encouraged me to compose the above, despite the assurances >> of Jim S that the 'abut*' usage is by now well entrenched. >> >> Fred From rrosebru@mta.ca Fri Aug 22 09:35:35 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 22 Aug 2008 09:35:35 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KWVrN-0005vR-CH for categories-list@mta.ca; Fri, 22 Aug 2008 09:35:21 -0300 From: Robert L Knighten MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Date: Thu, 21 Aug 2008 13:23:54 -0700 To: categories Subject: categories: Re: abutment = aboutement? Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 47 To add to the chatter on this topic I'll point to the bilingual Grothendieck-Serre Correspondence (Pierre Colmez, Jean-Pierre Serre, eds., Catriona Maclean, tranlater. AMS/SMF, 2004). The first appearance of "abutment" of a spectral sequence appears on p. 21 as the translation of Serre's "au bout" (quotation marks in the text.) On p. 26 Grothendieck uses "l'aboutissement" (in quotation marks) which is also translated as "abutment". Aboutissement and the translation abutment, without quotation marks, then appear on occasion throughout the remainder of the text. -- Bob -- Robert L. Knighten RLK@knighten.org From rrosebru@mta.ca Fri Aug 22 09:37:22 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 22 Aug 2008 09:37:22 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KWVsn-00063p-SD for categories-list@mta.ca; Fri, 22 Aug 2008 09:36:49 -0300 Date: Fri, 22 Aug 2008 00:04:53 -0400 From: "Fred E.J. Linton" Subject: categories: Re: abutment = aboutement? 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: 48 A few gentle corrections, if I may, and a comment. = First, a recurrent typo: *never* "abuter" -- only "abouter". Next, when I wrote > French "abouter" is a verb, derived from "bout", whose meaning is > 'to join (or to place) end to end' I omitted what I (evidently incorrectly) thought went without saying; better would have been to include it, so: > 'to join (or to place) [two things] end to end' (thus "abouter" can be used for the placement of two successive = spans of a bridge, as in Vaughan's illustration, with the special dedicated support where the two spans 'abut' being, obviously, an 'abutment'; but one would not ["abouter" a path]). And then, I omitted to mention that the verb "aboutir" is *intransitive* -- it does *not* accept any direct object. Thus it is linguistically impossible to ["aboutir" a path]: but one *may* say of a path that it "aboutit" *at* a certain point, or *in* a certain set, or ... . [There is a reflexive cognate of "aboutir" -- "s'aboutir" -- used in gardening terminology to mean 'to bud' or 'to be covered with buds', but this usage surerely serves only as a red herring if one wants = to understand "aboutir" proper.] Finally, to make peace with Jim S and Mike B: I in no way intend what I've written (initially just privately, first to Mike, and = then to Eduardo) to dictate new terminology in place of established = spectral sequence usage. And I very much appreciate Jim's having shared = his mental 'abutting' vision for that usage. And yet, remembering the triples/monads transition, I wonder whether a similar transition may not yet take place as regards "aboutissement", etc. Cheers, -- Fred [PS: As not all mail-readers render what are known as HTML named entities= = correctly, let me just add that, where a reader may see an 'agrave' betwe= en an ampersand and a semicolon, I had intended an "a" with 'accent grave'; my similarly placed 'eacute' was meant to show as "e" with 'acute accent'= =2E Apologies to all those whose mail-readers garble these. -- Fred] --- = ------ Original Message ------ Received: Thu, 21 Aug 2008 08:56:30 AM EDT From: "Eduardo J. Dubuc" To: Categories list Subject: categories: Re: abutment =3D aboutement? > Fred Linton has in this mail enlighten us all about the meaning in fren= ch of > the words "aboutissement" and "aboutement". Quite different meanings. > = > (*) It seems that "aboutir" means more or less "to arrive" or "to finis= h" > "come to the end" etc, > = > while "abuter" means 'to join end to end', .... From rrosebru@mta.ca Sat Aug 23 09:38:14 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 23 Aug 2008 09:38:14 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KWsLt-0001Jw-RE for categories-list@mta.ca; Sat, 23 Aug 2008 09:36:21 -0300 To: "Categories list" Content-Type: text/plain; charset="utf-8" Date: Fri, 22 Aug 2008 09:30:05 -0400 Subject: categories: Re: abutment = aboutement? From: wlawvere@buffalo.edu Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 49 =20 There are common words, rarely used in a technical sense, that however may be useful as explanatory marginal alternatives of foreign words. The "ab...ment d.. " being discussed seems to be explained by "goal", as in f is the goal of F where F is the Fourer series of a function f (which leaves to particular investigation the question of actual convergence). In turn "goal" can be helpfully explained as=20 purpose a concept that academic discussions should not forget. Bill On Thu 08/21/08 3:18 PM , Vaughan Pratt pratt@cs.stanford.edu sent: > Meanwhile I count eight occurrences of "abut" and > "abutment" in the (36kilobyte!) main Wikipedia article on spectral sequen= ces (there are a > dozen separate much shorter articles on particular spectral sequences, > along with a 15 kB article on derived categories). >=20 > On the other hand the algebra and geometry articles of the 1987 > Britannica Macropaedia both prefer the term "limit" for what a > spectralsequence converges to, in respectively Peter Hilton's contributio= n > "Other aspects of homological algebra" to the algebra article, > and thegeometry article's section on algebraic topology. >=20 > Since Wikipedia seems to be trumping Britannica these days, and no one > here has objected to established usage in mathematics trumping > linguistic suitability, the precise distance of "abutment" from > theoptimal English cognate for "aboutissement" would appear to be > academic,an epithet reflecting the outside world's perception that raisin= g moot > points is in our job description. >=20 > Vaughan >=20 > >> Thanks to Eduardo D and Vaughan P and Michel > H for their misgivings,>> which encouraged me to compose the above, > despite the assurances>> of Jim S that the 'abut*' usage is by now > well entrenched.>> > >> Fred >=20 >=20 >=20 >=20 >=20 >=20 From rrosebru@mta.ca Sat Aug 23 09:38:14 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 23 Aug 2008 09:38:14 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KWsL4-0001GI-B7 for categories-list@mta.ca; Sat, 23 Aug 2008 09:35:30 -0300 Date: Fri, 22 Aug 2008 09:22:08 -0400 From: jim stasheff MIME-Version: 1.0 To: Categories list Subject: categories: Re: abutment = aboutement? 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: 50 Thaks, Vaughan That reraises a question I implied earlier limit for the E_\infty term is appropriate but that is the graded of what I dimly recall the SS abuts to - the ungraded, e.g. H^(E) in the Serre-Leray SS anyone confirm that? jim Pratt wrote: > Meanwhile I count eight occurrences of "abut" and "abutment" in the (36 > kilobyte!) main Wikipedia article on spectral sequences (there are a > dozen separate much shorter articles on particular spectral sequences, > along with a 15 kB article on derived categories). > > On the other hand the algebra and geometry articles of the 1987 > Britannica Macropaedia both prefer the term "limit" for what a spectral > sequence converges to, in respectively Peter Hilton's contribution > "Other aspects of homological algebra" to the algebra article, and the > geometry article's section on algebraic topology. > > Since Wikipedia seems to be trumping Britannica these days, and no one > here has objected to established usage in mathematics trumping > linguistic suitability, the precise distance of "abutment" from the > optimal English cognate for "aboutissement" would appear to be academic, > an epithet reflecting the outside world's perception that raising moot > points is in our job description. > > Vaughan > >>> Thanks to Eduardo D and Vaughan P and Michel H for their misgivings, >>> which encouraged me to compose the above, despite the assurances >>> of Jim S that the 'abut*' usage is by now well entrenched. >>> >>> Fred > > From rrosebru@mta.ca Sat Aug 23 09:39:49 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 23 Aug 2008 09:39:49 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KWsP6-0001RX-Vk for categories-list@mta.ca; Sat, 23 Aug 2008 09:39:41 -0300 Date: Fri, 22 Aug 2008 13:46:06 -0300 From: "Eduardo J. Dubuc" MIME-Version: 1.0 To: categories Subject: categories: Re: abutment = aboutement? Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 51 [ Note from moderator: It is time to close this thread. Further discussion of the linguistic part of Michael's query should happen away from the list. Thanks to contributors. ] Michael Barr wrote: > I am NOT about to change a word that has apparently existed for over 50 > years just because it is not particularly meaningful. "aboutissement" existed for over 50 years of course, but the translation "abutment" is, (at least in the references given in these postings, including the translation of the grothendieck-serre correspondence) much more recent, it seem that all of its occurrences are from the second millennium, or very close. On the other hand, we have learned in these postings that in the 1980's and before, the word "limit" (and not "abutment') was used for what a spectral sequence converges to. An authority as Peter Hilton used "limit". There are a lot of well established words in mathematics which are not particularly meaningful, but the difference with "abutment" is precisely that they are well established. From rrosebru@mta.ca Sat Aug 23 09:44:31 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 23 Aug 2008 09:44:31 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KWsTL-0001fT-WB for categories-list@mta.ca; Sat, 23 Aug 2008 09:44:04 -0300 From: peasthope@shaw.ca Date: Fri, 22 Aug 2008 17:19:29 -0700 Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 To: categories@mta.ca Subject: categories: uniqueness of a map object Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 52 L&S page 314 Ex. 1, "... uniqueness proposition to the=20 effect that if M1, e1 and M2, e2 both serve as map objects=20 with evaluation map for T -> Y, then there is a unique=20 isomorphism between them which is compatible with the=20 evaluation structures."=20 Would this diagram be aiming in the right direction? http://carnot.yi.org/MapObjectUniqueness.jpg Thanks ... Peter E. --=20 http://members.shaw.ca/peasthope/ http://carnot.yi.org/ =3D http://carnot.pathology.ubc.ca/ From rrosebru@mta.ca Sat Aug 23 16:27:48 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 23 Aug 2008 16:27:48 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KWyki-0003kD-9p for categories-list@mta.ca; Sat, 23 Aug 2008 16:26:24 -0300 Date: Sat, 23 Aug 2008 09:17:55 -0400 (EDT) From: Michael Barr To: Categories list Subject: categories: Final post 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: 53 Karl Hofmann assures me that the domain of \beta^{p,q} is indeed E^{p,q}_\infty, which is the important point. As for the word used, it is unimportant since that approach to spectral sequences seems to have gone by the boards. Thanks to Mike Mislove for posing this question to Karl. And now maybe the moderator can shut down this thread. Michael From rrosebru@mta.ca Sun Aug 24 16:06:44 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 24 Aug 2008 16:06:44 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KXKsR-0003Lt-98 for categories-list@mta.ca; Sun, 24 Aug 2008 16:03:51 -0300 Date: Sat, 23 Aug 2008 22:06:19 -0400 From: "Fred E.J. Linton" To: Subject: categories: RIP: Henri Cartan, 1904-2008 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: 54 An obituary notice for Henri Cartan just came my way. It seemed appropriate to share portions of it here. Sadly, -- Fred ------ Extracts from Original Message ------ Received: Sat, 23 Aug 2008 05:48:37 AM EDT From: The Electronic IMU Newsletter =2E.. Henri Cartan (1904-2008) The world has lost one of the greatest scientists of the twentieth century. Henri Cartan, a legendary figure in mathematics, died in Paris on 13 August, at the age of 104 years. The son of the great mathematician Elie Cartan, his contributions to mathematics were fundamental, from several complex variables to algebraic topology and homological algebra. A member of the Bourbaki group, his participation in the rejuvenation of the French mathematical school was essential, in particular through his seminar held at the =C9cole Normale Sup=E9rieure. His roles as teacher and mentor= were also exceptional, and were felt well beyond national boundaries. =2E.. An interview of Henri Cartan conducted in March, 1999, and published in the Notices of the American Mathematical Society, may be found at http://www.ams.org/notices/199907/fea-cartan.pdf On-line obituaries may be found at http://www.zeit.de/online/2008/34/henri-cartan-nachruf Memorial Web sites for Henri Cartan French Academie des Sciences: http://www.academie-sciences.fr/membres/C/Cartan_Henri.htm Soci=E9t=E9 Math=E9matique de France: http://smf.emath.fr/en/VieSociete/Rencontres/JourneeCartan/NoticeCartan.h= tml 100th issue of the Gazette des Math=E9maticiens, the news publication of Soci=E9t=E9 Math=E9matique de France, carried some tributes to Cartan, tw= o of which are reproduced in September 2004 issue of the European Mathematical Society Newsletter http://www.emis.de/newsletter/archive_contents.html#nl_53 =2E.. From rrosebru@mta.ca Mon Aug 25 10:12:48 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 25 Aug 2008 10:12:48 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KXbpY-00007f-Ln for categories-list@mta.ca; Mon, 25 Aug 2008 10:10:00 -0300 Date: Sun, 24 Aug 2008 19:51:08 -0400 (EDT) From: Michael Barr To: Categories list Subject: categories: Asking for more trouble 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: 55 G is now talking about locally compact spaces and there are two phrases I have not seen. One is "relatively compact". I assume this is the same as what I call "conditionally compact", i.e. having compact closure. The other is "denombrable". The context is "Suppose that the locally compact space $X$ is denombrable a l'infini". Does it mean first countable? I don't want to start a discussion what bad terms these are, I just want to know what they mean. Michael From rrosebru@mta.ca Mon Aug 25 13:25:19 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 25 Aug 2008 13:25:19 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KXeqF-0002N6-HD for categories-list@mta.ca; Mon, 25 Aug 2008 13:22:55 -0300 Date: Mon, 25 Aug 2008 16:47:41 +0200 (CEST) Subject: categories: Re: Asking for more trouble From: Johannes.Huebschmann@math.univ-lille1.fr To: "Categories list" MIME-Version: 1.0 Content-Type: text/plain;charset=utf-8 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 56 These are Bourbaki terms. Denombrable: countable denombrable a l'infini: countable at infinity (the point at infinity of the Alexandroff compactification of a locally compact space has a countable neighborhood base) relativement compact: relatively compact (having compact closure) Johannes > G is now talking about locally compact spaces and there are two phrases= I > have not seen. One is "relatively compact". I assume this is the same= as > what I call "conditionally compact", i.e. having compact closure. The > other is "denombrable". The context is "Suppose that the locally compa= ct > space $X$ is denombrable a l'infini". Does it mean first countable? > > I don't want to start a discussion what bad terms these are, I just wan= t > to know what they mean. > > Michael > > > From rrosebru@mta.ca Mon Aug 25 13:25:19 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 25 Aug 2008 13:25:19 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KXeqp-0002Pn-Hh for categories-list@mta.ca; Mon, 25 Aug 2008 13:23:31 -0300 Date: Mon, 25 Aug 2008 11:46:01 -0400 (EDT) From: Robert Seely To: Categories List Subject: categories: Octoberfest 08 - some details MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 57 [ Webpage: http://www.math.mcgill.ca/triples/octoberfest08.html ] Centre de Recherche en Theorie des Categories -- Montreal -- Category Theory Research Center -------------------------------------------------------------------------- Category Theory OctoberFest Concordia University, Montreal Saturday - Sunday, October 4 - 5, 2008 -------------------------------------------------------------------------- We invite you to join us in Montreal next October for a weekend meeting in Category Theory, the "not-quite-annual" OctoberFest. As has been the tradition with these meetings, we invite talks from all participants. If you wish to give a talk, send your request along with a short abstract (well before the end of September please) to Robert Seely at the address below. The final schedule of talks will be handed out at the meeting, but a provisional schedule should be available on the webpage, if possible, before the meeting. We will meet in Concordia's Faubourg building ("FG" on the Concordia map) on Saturday morning, October 4th. The FG building is located at 1616 St. Catherine W., corner of St Mathieu (see the Campus map). Lectures will be in room FGB040 in the basement of the FG building. To access this, the easiest way is via the St Mathieu entrance, then go two floors down. We'll provide more explicit instructions on the website soon, and we'll post signs to help on the day. Coffee will be available from 8:30 am. The first talk will be at 9:00. Information re parking: Concordia has an indoor parking garage in the LB (Library) building, 1436 Mackay, near the site of the talks - the daily rate on weekends is only $6. There will be a very modest registration fee to cover refreshments and sundries (details to be specified later). Registration will take place during the morning, before the first talk and during the first coffee break. We plan to arrange a buffet lunch on Saturday at a local Indian restaurant (participants responsable for their own costs). Some recommendations for dinner will be available, but participants are officially on their own(!) in the evening. Please let us know if you intend to join us by sending a short email (before the end of September if possible) to Robert Seely. Below you will find a list of hotels and tourist rooms close to Concordia. We've provided links (on the webpage) to those that ought to be the most suitable. Please note that October is a popular month for visiting Montreal, and early booking is recommended. If you have any further questions, please contact one of us. -------------------------------------------------------------------------- Michael Barr [barr@math.mcgill.ca] Robert Raphael [raphael@alcor.concordia.ca] Robert Seely [rags@math.mcgill.ca] -------------------------------------------------------------------------- Other information to be found on the webpage: Weather: It can be warm, cold, wet, dry ... anything is possible in October. We suggest you check the forecast for the Montreal area Map of the Concordia campus area Parking General map of the area around Concordia -------------------------------------------------------------------------- Hotels: (check the website for links to the closest ones) * Le Chateau Versailles, 1808 Sherbrooke W, 514-933-8111 (1-888-933-8111) * Le Meridien Versailles, 1659 Sherbrooke W, 514-933-8111 (1-800-543-4300) * Hotel du Fort, 1390, rue du Fort, 514-938-8333 (1-800-565-6333) * Hotel Europa Best Western, 1240 rue Drummond, 514-866-6492 (1-800-361-3000) * L'Appartement*, 455 Sherbrooke W, 514-284-3634 * Howard Johnson Plaza*, 475 Sherbrooke W, 514-842-3961 (1-800-842-3961). * Mariott Courtyard*, 410 Sherbrooke W, 514-844-8851 * Holiday Inn*, 420 Sherbrooke W, 514-842-6111 * Hotel du Parc*, 3625 Park Ave, 514-288-6666 Tourist Rooms: (Prices vary depending on choice of single/double room, private/shared bathroom.) * Manoir Ambrose, 3422 Stanley, 514-288-6922, (excellent) * Armor***, 157 Sherbrooke E, 514-285-0140 * Bienvenue B&B*** 3950 Laval, 514-844-5897 (1-800-227-5897) * Casa Bella*, 258 Sherbrooke W, 514-849-2777 * Centre Ville B&B**, 3458 Laval, 514-289-9749 * Pierre***, 169 Sherbrooke E, 514-288-8519 *15 minute walk from Concordia **30 minute walk from Concordia ***45 minute walk from Concordia (Note that using the Metro or bus makes the trip shorter) -- From rrosebru@mta.ca Mon Aug 25 13:25:19 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 25 Aug 2008 13:25:19 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KXerT-0002Tf-6w for categories-list@mta.ca; Mon, 25 Aug 2008 13:24:11 -0300 Date: Mon, 25 Aug 2008 09:08:38 -0700 From: Toby Bartels To: Categories list Subject: categories: Re: Asking for more trouble 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: 58 Michael Barr wrote in part: >G is now talking about locally compact spaces and there are two phrases I >have not seen. One is "relatively compact". I assume this is the same as >what I call "conditionally compact", i.e. having compact closure. Depending on the author, "relatively compact" can mean either having compact closure, or having *any* compact superset. In a Hausdorff space, these are equivalent. --Toby From rrosebru@mta.ca Mon Aug 25 13:25:19 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 25 Aug 2008 13:25:19 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KXerT-0002Tf-6w for categories-list@mta.ca; Mon, 25 Aug 2008 13:24:11 -0300 Date: Mon, 25 Aug 2008 09:08:38 -0700 From: Toby Bartels To: Categories list Subject: categories: Re: Asking for more trouble 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: 59 Michael Barr wrote in part: >G is now talking about locally compact spaces and there are two phrases I >have not seen. One is "relatively compact". I assume this is the same as >what I call "conditionally compact", i.e. having compact closure. Depending on the author, "relatively compact" can mean either having compact closure, or having *any* compact superset. In a Hausdorff space, these are equivalent. --Toby From rrosebru@mta.ca Mon Aug 25 17:10:27 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 25 Aug 2008 17:10:27 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KXiLu-0005Tv-EY for categories-list@mta.ca; Mon, 25 Aug 2008 17:07:50 -0300 Date: Mon, 25 Aug 2008 13:27:13 -0400 (EDT) From: Michael Barr To: Categories list Subject: categories: Resolution MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 60 Thanks to Jonathan Chiche and Johannes Huebschman for the answer to my question. First off, according to the online Encyclopedia of Mathematics, relatively compact means having compact closure (I had called that conditionally compact; neither term is very evocative). Now to denombrable a l'infini, first Johannes wrote that it meant that the one point compactification had a countable basis at the point at infinity. Then Jonathan pointed to a '57 paper of M. Zisman that actually defined it to mean \sigma-compact. In the context of locally compact spaces, the two definitions are easily seen to be equivalent! Since \sigma-compact seems to be widely used, I will go with that. And now let us break off this thread. Michael From rrosebru@mta.ca Tue Aug 26 20:04:14 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 26 Aug 2008 20:04:14 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KY7Xk-0000sL-Ln for categories-list@mta.ca; Tue, 26 Aug 2008 20:01:44 -0300 Date: Tue, 26 Aug 2008 16:07:58 -0400 (EDT) From: Bill Lawvere To: categories@mta.ca Subject: categories: Re: KT Chen's smooth CCC, a correction MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 61 Dear Jim and colleagues, By urging the study of the good geometrical ideas and constructions of Chen and Froelicher, as well as of Bott, Brown, Hurewicz, Mostow, Spanier, Steenrod, I am of course not advocating the preferential resurrection of the particular categories they tentatively devised to contain the constructions. Rather, recall as an analogy the proliferation of homology theories 60 years ago; it called for the Eilenberg-Steenrod axioms to unite them. Similarly, the proliferation of such smooth categories 45 years ago would have needed a unification. Programs like SDG and Axiomatic Cohesion have been aiming toward such a unification. The Eilenberg-Steenrod program required, above all, the functorality with respect to general maps; in that way it provided tools to construct even those cohomologies (such as compact support and L2 theories) that are less functorial. The pioneers like Chen recognized that the constructions of interest (such as a smooth space of piecewise smooth paths or a smooth classifying space for a Lie group) should take place in a category with reasonable function spaces. They also realized, like Hurewicz in his 1949 Princeton lectures, that the primary geometric structure of the spaces in such categories must be given by figures and incidence relations (with the algebra of functions being determined by naturality from that, rather than conversely as had been the 'default' paradigm in 'general' topology, where the algebra of Sierpinski-valued functions had misleadingly seemed more basic than Frechet-shaped figures.) I have discussed this aspect in my Palermo paper on Volterra (2000). The second aspect of the default paradigm, which those same pioneers seemingly failed to take fully into account, is repudiated in the first lines of Eilenberg & Zilber's 1950 paper that introduced the key category of Simplicial Sets. Some important simplicial sets having only one point are needed (for example, to construct the classifying space of a group). Therefore. the concreteness idea (in the sense of Kurosh) is misguided here, at least if taken to mean that the very special figure shape 1 is faithful on its own. That idea came of course from the need to establish the appropriate relation to a base category U such as Cantorian abstract sets, but that is achieved by enriching E in U via E(X,Y) = p(Y^X), without the need for faithfulness of p:E->U; this continues to make sense if E consists not of mere cohesive spaces but of spaces with dynamical actions or Dubuc germs, etcetera, even though then p itself extracts only equilibrium points. The case of simplicial sets illustrates that whether 1 is faithful just among given figure shapes alone has little bearing on whether that is true for a category of spaces that consist of figures of those shapes. Naturally with special sites and special spaces one can get special results: for example, the purpose of map spaces is to permit representing a functional as a map, and in some cases the structure of such a map reduces to a mere property of the underlying point map. Such a result, in my Diagonal Arguments paper (TAC Reprints) was exemplified by both smooth and recursive contexts; in the latter context Phil Mulry (in his 1980 Buffalo thesis) developed the Banach-Mazur-Ersov conception of recursive functionals in a way that permits shaded degrees of nonrecursivity in domains of partial maps, yet as well permits collapse to a 'concrete' quasitopos for comparison with classical constructions. Grothendieck did fully assimilate the need to repudiate the second aspect (as indeed already Galois had done implicitly; note that in the category of schemes over a field the terminal object does not represent a faithful functor to the abstract U). Therefore Grothendieck advocated that to any geometric situations there are, above all, toposes associated, so that in particular the meaningful comparisons between geometric situations start with comparing their toposes. A Grothendieck topos is a quasitopos that satisfies the additional simplifying axiom: All monomorphisms are equalizers. A host of useful exactness properties follows, such as: (*)All epimonos are invertible. The categories relevant to analysis and geometry can be nicely and fully embedded in categories satisfying the property (*). That claim arouses instant suspicion among those who are still in the spell of the default paradigm; for that reason it may take a while for the above-mentioned 45-year-old proliferation of geometrical category-ideas to become recognized as fragments of one single theory. There is still a great deal to be done in continuing K.T. Chen's application of such mathematical categories to the calculus of variations and in developing applications to other aspects of engineering physics. These achievements will require that students persist in the scientific method of alert participation, like guerilla fighters pursuing the laborious and cunning traversal of a treacherous jungle swamp. For in the maze of informative 21st century conferences and internet sites there lurk fickle pedias and beckening bistros which, like the mythical black holes, often regurgitate information as buzzwords and disinformation. Bill On Sun, 17 Aug 2008, jim stasheff wrote: > Bill, > > Happy to see you contributing to the renaissance in interest in > Chen's work. > > It would be good to post your msg to the n-category cafe blog > whee there's been an intense discussion of `smooth spaces' i various > incarnaitons. > > jim > > http://golem.ph.utexas.edu/category/2008/05/convenient_categories_of_smoot.html > > wlawvere@buffalo.edu wrote: >> In my review of Anders Kock's Synthetic >> Differential Geometry, Second Edition, >> there is a wrong statement that I want to correct. >> (This was in the SIAM REVIEW, vol. 49, No.2 >> pp 349-350). The statement was that Chen's >> category does not include the representability >> of smooth function spaces. But from his paper >> In Springer Lecture Notes in Mathematics,vol >> 1174, pp 38-42 it is clear that it does. I thank Anders >> for pointing out this slip. >> >> This is a good opportunity to emphasize that >> the works of KT Chen and of Alfred Frolicher >> (that were referred to in the beginning of the >> above review) contain several contributions >> of value both to applications and to more >> topos-theoretic formulations. For example, >> Frolicher's use of Lemmas by Boman and others >> reveals how little of the specific parameter "smooth" >> needs to be given to the very general machinery of >> adjoint functors and abstact sets in order to obtain >> smooth infinite dimensional spaces of all kinds. >> (Namely a suitable topos of actions by only unary >> operations on the line is fully embedded >> in the desired topos in such a way that the algebraic >> theory of n-ary operations that naturally exist in >> the small one determines the whole algebraic category whose >> sheaves include the large one.) >> And Chen's smooth space of piecewise-smooth >> curves can surely be further applied, as can his >> special use of convex models for plots. >> >> Bill Lawvere >> >> >> >> > > > > > From rrosebru@mta.ca Wed Aug 27 16:42:48 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 27 Aug 2008 16:42:48 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KYQtA-00027z-J5 for categories-list@mta.ca; Wed, 27 Aug 2008 16:41:08 -0300 Date: Tue, 26 Aug 2008 21:11:30 -0400 From: "Fred E.J. Linton" To: Subject: categories: Re: Resolution 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: 62 Greetings Let's just hope none of this creates another situation like the one Sammy reported facing in a North African fish restaurant, where his menu offered, among other local delicacies, "Fried Pimp", the author evidently = having rendered the Arabic word for the fish in question, = actually a mackerel, first into French as "maquereau", = and thence into English as "pimp". = Rhymes with "shrimp" -- easier to type than "mackerel" -- so why not? Cheers, -- Fred ------ Original Message ------ Received: Mon, 25 Aug 2008 04:13:56 PM EDT From: Michael Barr To: Categories list Subject: categories: Resolution > Thanks to Jonathan Chiche and Johannes Huebschman for the answer to my > question. First off, according to the online Encyclopedia of Mathemati= cs, > relatively compact means having compact closure (I had called that > conditionally compact; neither term is very evocative). > = > Now to denombrable a l'infini, first Johannes wrote that it meant that = the > one point compactification had a countable basis at the point at infini= ty. > Then Jonathan pointed to a '57 paper of M. Zisman that actually defined= it > to mean \sigma-compact. In the context of locally compact spaces, the = two > definitions are easily seen to be equivalent! Since \sigma-compact see= ms > to be widely used, I will go with that. > = > And now let us break off this thread. > = > Michael > = > = > = From rrosebru@mta.ca Wed Aug 27 16:43:20 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 27 Aug 2008 16:43:20 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KYQvB-0002FL-40 for categories-list@mta.ca; Wed, 27 Aug 2008 16:43:13 -0300 From: "R Brown" To: Subject: categories: Re: KT Chen's smooth CCC, a correction Date: Wed, 27 Aug 2008 11:51:31 +0100 MIME-Version: 1.0 Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 63 Dear Bill and Colleagues,=20 I would like to explain my own interest in function spaces and function = objects since it has a different origin to what Bill explains and a = different direction which could be of interest for comment and = investigation.=20 Michael Barratt suggested to me in 1960 the problem of calculating the = homotopy type of the space X^Y by induction on the Postnikov system of = X, in contrast to Michael's own work on Track Groups, where he used a = homology decomposition of Y, and using Whitney's tube systems gave = explicit description of some group extensions in examples of the = Barratt(-Puppe) exact sequence. (amazing!!?)=20 Now the first Postnikov invariant in its simplest form is a Sq^2 but the = extension is described by a Sq^1. How did the one transform into the = other? Clue: the Cartan formula for Sq^2 on a product. How did a product = get into the act? Answer: the evaluation map!=20 Trying to write all this down led to using a number of `exponential = laws' in spaces, spaces with base point, simplicial sets, pointed = simplicial sets, chain complexes, simplicial abelian groups, etc. So it = was dinned into me that an exponential law depended on the product as = well as the function object. So why not try the known weak product for = topological spaces? Surprise, surprise, it all worked, and was part 1 of = my thesis, submitted 1961, with a sketchy account of what we now call = monoidal closed categories, exemplified, but not developed in general = terms.=20 Subsequent work with Philip Higgins has continued to use monoidal closed = categories in algebraic topology. Indeed the category of crossed = complexes is cartesian closed, but the homotopy theory one wants is = given by the (different) monoidal closed structure. So the category of = filtered spaces is usefully enriched over this monoidal closed = category.=20 My question is then: what is the potential influence of this need for = monoidal closed? It clearly does not lead to topos theory as such. It = does lead to the possibility of some not previously available = calculations, even of nonabelian homotopical invariants, and is relevant = to the study of local-to-global problems. (I first heard these words = from Dick Swan in connection with sheaf theory.) But in this work = cubical sets became essential, for ease of discussing subdivision, = multiple compositions, and homotopies, and here the monoidal closed = structure is crucial. Kan's initial cubical work was neglected in favour = of the (convenient in many ways) cartesian closed category of simplicial = sets.=20 One specific problem for me was a general notion of symmetry (naively, = and using buzz words (!), higher order groupoids should yield higher = order notions of symmetry!). In a cartesian closed category C we have = for a specific object x not only Aut(x), the isomorphisms of x, but also = AUT(x), the internal group object of automorphisms of x. This has been = developed for the topos of directed graphs, in John Shrimpton's thesis, = but actually the unaccomplished aim was to understand Grothendieck's = Teichmuller Groupoid, and his envisaged computations of this by gluing = or clutching procedures, but which needed topos theory, he claimed! When = I asked for any notes on this he just said nothing was written down, it = was all in his mind. Baffling! =20 In the monoidal case we can get only that END(x) is an internal monoid = wrt tensor. But in some cases we have a candidate for AUT(x), even if = `internal group' wrt tensor makes no sense. One example was worked out = with Nick Gilbert (published 1989). In this case there is a forgetful = functor U to a cartesian closed category, in this case Set, and you can = make up the rest. It worked in this dimension, relevant to homotopy = 3-types, but still did not lead by induction to even higher order = notions of symmetry. Pity!=20 My question is now: given this background, how should we match the = beautiful ideas and insights of Bill with what seem to be some monoidal = closed realities? Could this be important for geometry, and, better = still, even for analysis, and dynamics?=20 Ronnie From: Bill Lawvere To: categories@mta.ca Sent: Tuesday, 26 August, 2008 9:07:58 PM Subject: categories: Re: KT Chen's smooth CCC, a correction Dear Jim and colleagues, By urging the study of the good geometrical ideas and constructions of Chen and Froelicher, as well as of Bott, Brown, Hurewicz, Mostow, = Spanier, Steenrod, I am of course not advocating the preferential resurrection of the particular categories they tentatively devised to contain the constructions. ... From rrosebru@mta.ca Wed Aug 27 16:44:46 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 27 Aug 2008 16:44:46 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KYQwX-0002Kj-8H for categories-list@mta.ca; Wed, 27 Aug 2008 16:44:37 -0300 Date: Wed, 27 Aug 2008 14:23:46 -0400 (EDT) From: Bill Lawvere Reply-To: Regula Honegger To: categories@mta.ca Subject: categories: Re: KT Chen's smooth CCC, a correction MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 64 Dear Ronnie and Colleagues, Your comments are extremely interesting. Thank you very much for raising in so striking a manner the question of the relation between general monoidal structures and cartesian closed structures. Below are some observations which show, I think, that everybody should be interested in this relation because it is manyfold and fruitful. (1) While cartesian closed structures have the virtue of being unique, general monoidal closed structures have the virtue of not being unique. Thus, for example, the cartesian closed presheaf toposes (with their exactness properties and combinatorial truth object) often have a further monoidal closed structure given by Brian Day's convolution with respect to a pro-co-monoidal structure on the site. Cubical as well as simplicial sets have both cartesian and non-cartesian closed structures, and that is 'true', not merely 'convenient'. (2) Another category having both cartesian and non-cartesian monoidal structures is the real interval from zero to infinity with 'x dominates y' as the morphism from x to y. (Actually, this category is derived by collapsing a natural topos of dynamical systems in 'Taking categories seriously' TAC Reprints.) Categories enriched with respect to the non-cartesian structure here (see 'Metric Spaces' TAC reprints) arise every day in analysis and the rich insights of enrichment theory (Functor categories, bi-module composition, free categories, etcetera) should be systematically applied to the advance of analysis and geometry, while on the other hand metric examples inspire further developments of enrichment theory. Cauchy (who never worked on idempotent splitting in ordinary categories and additive categories in the way that Freyd and Karoubi did) does not deserve to have his name brandished as a joke to scare one's uncomprehending colleagues in analysis. The kind of completeness that is inspired by two-sided intervals (unlike the one-sided intervals inaccurately alluded to in common discussions of 'density') indeed reduces to the one attributed to Cauchy in the particular example of Metric Spaces. The author hoped that observation would contribute to the advance of analysis and the development of enrichment theory, not to the supply of buzzwords. In fact, there is an insufficiently known branch of analysis called 'Idempotent Analysis', which deals largely with composition of bi-modules, or more precisely, with the relation between the two closed structures on the infinite interval. Of course, that monoidal category is isomorphic to the unit interval under multiplication (still cartesian closed too) which induces many of the relations between probablility and entropy. (3) Perhaps the most common relation between non-cartesian monoidal categories and cartesian categories arises when a structure such as vector space is interpreted in a cohesive background. I am sticking to my story that cohesive backgrounds are basically cartesian closed, due to the ubiquitous role of diagonal maps and also due to the fact that, for example, bornological vector spaces have an obvious monoidal closed structure, whereas topological vector spaces have none. The rumor that topological vector spaces might have a tensor with an adjoint hom is part of the disinformation that makes functional analysis look more difficult than it is. A more accurate account of the relation between non-Mackey convergence and closed structure can be found in C. Houzel's paper on Grauert finiteness, Mathematische Annalen, vol. 205, 1973, 13-54: essentially, the topological categories are merely enriched in the genuinely monoidal closed bornological ones. Similarly, the idea that not all dual spaces are complete seems to be based on a misguided generality in the notion of Cauchy nets (they should be bounded). (4) Although pointed spaces are somewhat entrenched in algebraic topology, there is an improvement suggested by your own work, Ronnie. Consider the category whose objects are arrows S ---> E where E is a space (object of a cartesian closed cohesive background category) and S is a discrete space. This category is even a topos if the category of E's was, as is the larger category of arrows between general pairs of spaces. The first category is actually an adjoint retract of the second, correcting the discontinuity that arises from the traditional limitation S = 1. Intuitively, in the case where the pair of spaces is a subspace inclusion, the adjoint collapses the subspace to a point if the subspace is connected, but if it is not connected, does not artificially merge its components. There are many applications of this corrected construction of the space which results from 'neglecting' a subspace, both in algebraic topology and in functional analysis, too numerous to discuss here. Bill On Wed, 27 Aug 2008, R Brown wrote: > Dear Bill and Colleagues, > > I would like to explain my own interest in function spaces and function > objects since it has a different origin to what Bill explains and a > different direction which could be of interest for comment and > investigation. ... From rrosebru@mta.ca Thu Aug 28 15:35:42 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 28 Aug 2008 15:35:42 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KYmIm-0005JX-2t for categories-list@mta.ca; Thu, 28 Aug 2008 15:33:00 -0300 From: "R Brown" To: Subject: categories: Smooth categories etc Date: Thu, 28 Aug 2008 15:03:26 +0100 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: 65 Dear Bill and Colleagues,=20 In reply here only to Bill (4), but great interest in the other = comments, I entirely agree with the many pointed approach, which was = published in my 1967 Proc LMS article. This is relevant also to higher = homotopy theory. If you define \pi_n(X,P) where P is now a *set* of base = points, then you see this has to have the structure of module over = \pi_1(X,P). As an example of its use, consider the map=20 S^n \vee [0,1] \to S^n \vee S^1 which identifies 0 and 1, where the S^n is stuck to [0,1] at 0. Clearly = \pi_n of the first space on the set of base points P consisting of 0 and = 1 is the free module on one generator over the indiscrete groupoid = \I=3D\pi_1([0,1],P). The main theorem of Higgins and RB implies that = \pi_n of the second space at 0 is the free module on one generator over = the infinite cyclic group, which itself is obtained from \I by = identifying 0 and 1 in the category of groupoids.=20 I know this result can also be obtained using covering space arguments = and homology, but I got into groupoids by trying to avoid this detour to = covering spaces to describe the fundamental group of S^1. It is a = question of finding algebra which models geometry.=20 One of my joint papers, which calculated (using vKT for crossed modules) = some \pi_2(X,x) as a module, was rejected by one journal on the = grounds that the calculations were too elaborate when `the interest is = in the group and not the module'. So much for homological algebra!=20 There is a culture in algebraic topology which neglects the operations = of \pi_1; perhaps it seems an encumbrance when there is only one base = point, but Henry Whitehead commented in 1957 that the operations = fascinated the early workers in homotopy theory.=20 It is amusing to speculate what might be infinite loop space theory, or = little cube operads, if you allow many base points! Could it clear up = the subjects??!! Answers on a postcard please (joke).=20 A standard lesson in mathematics is that you should forget structure at = the latest possible moment. In homotopy theory low dimensional = identifications can and usually do affect higher dimensional homotopy = invariants. To try and cope with this, we need homotopical functors = which carry algebraic information in a range of dimensions, to model how = spaces are glued together. This has been the aim of the various Higher = Homotopy van Kampen theorems. To handle these algebraic structures one = needs category theory ; as one example, I am currently working on a = joint paper using fibred and cofibred categories to relate high and low = dimensional information on colimits and induced structures. =20 The hope also is that because of the wide interest in deformation, i.e. = homotopy, as a means of classification, these tools and methods will = have wider implications.=20 Ronnie Dear Ronnie and Colleagues, Your comments are extremely interesting. Thank you very much for = raising=20 in so striking a manner the question of the relation between general=20 monoidal structures and cartesian closed structures. Below are some observations which show, I think, that everybody should be interested in this relation because it is=20 manyfold and fruitful. (1) While cartesian closed structures have the virtue of being = unique,=20 general monoidal closed structures have the virtue of not being unique.=20 Thus, for example, the cartesian closed presheaf toposes (with their=20 exactness properties and combinatorial truth object) often have a = further=20 monoidal closed structure given by Brian Day's convolution with respect = to=20 a pro-co-monoidal structure on the site. Cubical as well as simplicial=20 sets have both cartesian and non-cartesian closed structures, and that = is=20 'true', not merely 'convenient'. (2) Another category having both cartesian and non-cartesian monoidal = structures is the real interval from zero to infinity with 'x dominates = y'=20 as the morphism from x to y. (Actually, this category is derived by=20 collapsing a natural topos of dynamical systems in 'Taking categories=20 seriously' TAC Reprints.) Categories enriched with respect to the=20 non-cartesian structure here (see 'Metric Spaces' TAC reprints) arise=20 every day in analysis and the rich insights of enrichment theory = (Functor=20 categories, bi-module composition, free categories, etcetera) should be=20 systematically applied to the advance of analysis and geometry, while on = the other hand metric examples inspire further developments of = enrichment=20 theory. Cauchy (who never worked on idempotent splitting in ordinary=20 categories and additive categories in the way that Freyd and Karoubi = did)=20 does not deserve to have his name brandished as a joke to scare one's=20 uncomprehending colleagues in analysis. The kind of completeness that is = inspired by two-sided intervals (unlike the one-sided intervals=20 inaccurately alluded to in common discussions of 'density') indeed = reduces=20 to the one attributed to Cauchy in the particular example of Metric=20 Spaces. The author hoped that observation would contribute to the = advance=20 of analysis and the development of enrichment theory, not to the supply = of=20 buzzwords. In fact, there is an insufficiently known branch of analysis called=20 'Idempotent Analysis', which deals largely with composition of = bi-modules,=20 or more precisely, with the relation between the two closed structures = on=20 the infinite interval. Of course, that monoidal category is isomorphic = to=20 the unit interval under multiplication (still cartesian closed too) = which=20 induces many of the relations between probablility and entropy. (3) Perhaps the most common relation between non-cartesian monoidal=20 categories and cartesian categories arises when a structure such as = vector=20 space is interpreted in a cohesive background. I am sticking to my story = that cohesive backgrounds are basically cartesian closed, due to the=20 ubiquitous role of diagonal maps and also due to the fact that, for=20 example, bornological vector spaces have an obvious monoidal closed=20 structure, whereas topological vector spaces have none. The rumor that=20 topological vector spaces might have a tensor with an adjoint hom is = part=20 of the disinformation that makes functional analysis look more difficult = than it is. A more accurate account of the relation between non-Mackey=20 convergence and closed structure can be found in C. Houzel's paper on=20 Grauert finiteness, Mathematische Annalen, vol. 205, 1973, 13-54: essentially, the topological categories are merely enriched in the=20 genuinely monoidal closed bornological ones. Similarly, the idea that = not=20 all dual spaces are complete seems to be based on a misguided generality = in the notion of Cauchy nets (they should be bounded). (4) Although pointed spaces are somewhat entrenched in algebraic=20 topology, there is an improvement suggested by your own work, Ronnie.=20 Consider the category whose objects are arrows S ---> E where E is a = space=20 (object of a cartesian closed cohesive background category) and S is a=20 discrete space. This category is even a topos if the category of E's = was,=20 as is the larger category of arrows between general pairs of spaces. The = first category is actually an adjoint retract of the second, correcting=20 the discontinuity that arises from the traditional limitation S =3D 1.=20 Intuitively, in the case where the pair of spaces is a subspace = inclusion,=20 the adjoint collapses the subspace to a point if the subspace is=20 connected, but if it is not connected, does not artificially merge its=20 components. There are many applications of this corrected construction = of=20 the space which results from 'neglecting' a subspace, both in algebraic=20 topology and in functional analysis, too numerous to discuss here. Bill From rrosebru@mta.ca Sat Aug 30 10:57:03 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 30 Aug 2008 10:57:03 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KZQuC-00023H-Ih for categories-list@mta.ca; Sat, 30 Aug 2008 10:54:20 -0300 Date: Sat, 30 Aug 2008 01:14:26 +0100 (BST) Subject: categories: Re: KT Chen's smooth CCC, a correction 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: RO X-Status: X-Keywords: X-UID: 66 On Tue, 26 Aug 2008, Bill Lawvere wrote: > fickle pedias and beckening bistros which, like the mythical > black holes, often regurgitate information as buzzwords and > disinformation. Disinformation is *deliberate* false information, false information *intended* to mislead. As I understand it, Bill's statement says, among other things, that disinformation often appears on the n-Category Cafe. I don't know whether Bill really meant to say this. I very much hope not. I can't think of a single instance where someone at the n-Category Cafe has intended to mislead. Best wishes, Tom On Tue, 26 Aug 2008, Bill Lawvere wrote: > Dear Jim and colleagues, > > By urging the study of the good geometrical ideas and constructions of > Chen and Froelicher, as well as of Bott, Brown, Hurewicz, Mostow, Spani= er, > Steenrod, I am of course not advocating the preferential resurrection o= f > the particular categories they tentatively devised to contain the > constructions. > > Rather, recall as an analogy the proliferation of homology theories 60 > years ago; it called for the Eilenberg-Steenrod axioms to unite them. > Similarly, the proliferation of such smooth categories 45 years ago wou= ld > have needed a unification. Programs like SDG and Axiomatic Cohesion hav= e > been aiming toward such a unification. > > The Eilenberg-Steenrod program required, above all, the functorality wi= th > respect to general maps; in that way it provided tools to construct ev= en > those cohomologies (such as compact support and L2 theories) that are > less functorial. > > The pioneers like Chen recognized that the constructions of interest (s= uch > as a smooth space of piecewise smooth paths or a smooth classifying spa= ce > for a Lie group) should take place in a category with reasonable functi= on > spaces. They also realized, like Hurewicz in his 1949 Princeton lecture= s, > that the primary geometric structure of the spaces in such categories m= ust > be given by figures and incidence relations (with the algebra of functi= ons > being determined by naturality from that, rather than conversely as had > been the 'default' paradigm in 'general' topology, where the algebra of > Sierpinski-valued functions had misleadingly seemed more basic than > Frechet-shaped figures.) I have discussed this aspect in my Palermo pap= er > on Volterra (2000). > > The second aspect of the default paradigm, which those same pioneers > seemingly failed to take fully into account, is repudiated in the first > lines of Eilenberg & Zilber's 1950 paper that introduced the key catego= ry > of Simplicial Sets. Some important simplicial sets having only one poin= t > are needed (for example, to construct the classifying space of a group)= . > Therefore. the concreteness idea (in the sense of Kurosh) is misguided > here, at least if taken to mean that the very special figure shape 1 is > faithful on its own. That idea came of course from the need to establis= h > the appropriate relation to a base category U such as Cantorian abstrac= t > sets, but that is achieved by enriching E in U via E(X,Y) =3D p(Y^X), > without the need for faithfulness of > p:E->U; this continues to make sense if E consists not of mere cohesiv= e > spaces but of spaces with dynamical actions or Dubuc germs, etcetera, e= ven > though then p itself extracts only equilibrium points. The case of > simplicial sets illustrates that whether 1 is faithful just among given > figure shapes alone has little bearing on whether that is true for a > category of spaces that consist of figures of those shapes. > > Naturally with special sites and special spaces one can get special > results: for example, the purpose of map spaces is to permit representi= ng > a functional as a map, and in some cases the structure of such a map > reduces to a mere property of the underlying point map. Such a result, = in > my Diagonal Arguments paper (TAC Reprints) was exemplified by both smoo= th > and recursive contexts; in the latter context Phil Mulry (in his 1980 > Buffalo thesis) developed the Banach-Mazur-Ersov conception of recursiv= e > functionals in a way that permits shaded degrees of nonrecursivity in > domains of partial maps, yet as well permits collapse to a 'concrete' > quasitopos for comparison with classical constructions. > > Grothendieck did fully assimilate the need to repudiate the second aspe= ct > (as indeed already Galois had done implicitly; note that in the categor= y > of schemes over a field the terminal object does not represent a faithf= ul > functor to the abstract U). Therefore Grothendieck advocated that to an= y > geometric situations there are, above all, toposes associated, so that = in > particular the meaningful comparisons between geometric situations star= t > with comparing their toposes. > > A Grothendieck topos is a quasitopos that satisfies the additional > simplifying axiom: > All monomorphisms are equalizers. > A host of useful exactness properties follows, such as: > (*)All epimonos are invertible. > The categories relevant to analysis and geometry can be nicely and full= y > embedded in categories satisfying the property (*). That claim arouses > instant suspicion among those who are still in the spell of the default > paradigm; for that reason it may take a while for the above-mentioned > 45-year-old proliferation of geometrical category-ideas to become > recognized as fragments of one single theory. > > There is still a great deal to be done in continuing > K.T. Chen's application of such mathematical categories to the calcul= us > of variations and in developing applications to other aspects of > engineering physics. These achievements will require that students pers= ist > in the scientific method of alert participation, like guerilla fighters > pursuing the laborious and cunning traversal of a treacherous jungle > swamp. For in the maze of informative 21st century conferences and > internet sites there lurk fickle pedias and beckening bistros which, li= ke > the mythical black holes, often regurgitate information as buzzwords an= d > disinformation. > > Bill > > > On Sun, 17 Aug 2008, jim stasheff wrote: > >> Bill, >> >> Happy to see you contributing to the renaissance in interest in >> Chen's work. >> >> It would be good to post your msg to the n-category cafe blog >> whee there's been an intense discussion of `smooth spaces' i various >> incarnaitons. >> >> jim >> >> http://golem.ph.utexas.edu/category/2008/05/convenient_categories_of_s= moot.html ----- The University of Glasgow, charity number SC004401 From rrosebru@mta.ca Sat Aug 30 10:57:04 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 30 Aug 2008 10:57:04 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KZQwd-0002CX-Iz for categories-list@mta.ca; Sat, 30 Aug 2008 10:56:51 -0300 Date: Sat, 30 Aug 2008 14:32:52 +0530 From: "chander pawar" To: categories@mta.ca Subject: categories: capturing regularity in languages 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: 67 hi, this is related to theory of formal languages. i would like to know if anyone has some references for , categories of formal languages, in particular "regular languages". how do we capture the notion of 'regularity' in categories? chander From rrosebru@mta.ca Sat Aug 30 10:58:12 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 30 Aug 2008 10:58:12 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KZQxo-0002Fv-A5 for categories-list@mta.ca; Sat, 30 Aug 2008 10:58:04 -0300 Date: Sat, 30 Aug 2008 10:34:32 +0200 (CEST) From: Johannes Huebschmann To: categories@mta.ca Subject: categories: Re: Resolution; "abutting to" 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: 68 Dear All This "situation" reminds me of the following: In 1976 Alexandrov was 80 years old. A. Dold was invited to contribute to the Festschrift, and so he did. Dold wrote his contribution in English and it got translated thereafter into Russian. A little while after the Festschrift had appeared Dold received a letter from a professional translator who had translated his paper from Russian into English. Johannes P. S. I have now found a reference for "spectral sequence abutting to": P. 187 of: J. C. Moore, Cartan's constructions, Colloque analyse et topologie, en l'honneur de Henri Cartan, Ast\'erisque 32--33 (1976), 173--221 On Tue, 26 Aug 2008, Fred E.J. Linton wrote: > Greetings > > Let's just hope none of this creates another situation > like the one Sammy reported facing in a North African > fish restaurant, where his menu offered, among other > local delicacies, "Fried Pimp", the author evidently > having rendered the Arabic word for the fish in question, > actually a mackerel, first into French as "maquereau", > and thence into English as "pimp". > > Rhymes with "shrimp" -- easier to type than "mackerel" -- > so why not? > > Cheers, -- Fred > > ------ Original Message ------ > Received: Mon, 25 Aug 2008 04:13:56 PM EDT > From: Michael Barr > To: Categories list > Subject: categories: Resolution > >> Thanks to Jonathan Chiche and Johannes Huebschman for the answer to my >> question. First off, according to the online Encyclopedia of Mathematics, >> relatively compact means having compact closure (I had called that >> conditionally compact; neither term is very evocative). >> >> Now to denombrable a l'infini, first Johannes wrote that it meant that the >> one point compactification had a countable basis at the point at infinity. >> Then Jonathan pointed to a '57 paper of M. Zisman that actually defined it >> to mean \sigma-compact. In the context of locally compact spaces, the two >> definitions are easily seen to be equivalent! Since \sigma-compact seems >> to be widely used, I will go with that. >> >> And now let us break off this thread. >> >> Michael >> >> >> > > > > > > From rrosebru@mta.ca Sun Aug 31 22:39:14 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 31 Aug 2008 22:39:14 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KZyLM-0002ul-6a for categories-list@mta.ca; Sun, 31 Aug 2008 22:36:36 -0300 From: "George Janelidze" To: Subject: categories: More CT2008 information will be in September Date: Mon, 1 Sep 2008 01:35:23 +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: 69 Dear Colleagues, This is just to say, that there will more information on CT2008 in September (since I have not sent in August, as I promised before). I apologize for the delay. George Janelidze