From MAILER-DAEMON Thu May 14 23:20:30 2009 Date: 14 May 2009 23:20:30 -0300 From: Mail System Internal Data Subject: DON'T DELETE THIS MESSAGE -- FOLDER INTERNAL DATA Message-ID: <1242354030@mta.ca> X-IMAP: 1238606430 0000000063 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 Wed Apr 1 13:52:46 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 01 Apr 2009 13:52:46 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lp3gA-0005xz-6L for categories-list@mta.ca; Wed, 01 Apr 2009 13:52:42 -0300 From: Thorsten Altenkirch To: Categories Mailing List Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit Mime-Version: 1.0 (Apple Message framework v930.3) Subject: categories: Where does the term monad come from? Date: Wed, 1 Apr 2009 12:24:46 +0100 Sender: categories@mta.ca Precedence: bulk Reply-To: Thorsten Altenkirch Message-Id: Status: O X-Status: X-Keywords: X-UID: 1 A question just came up at the Midland Graduate School (actually in the functional programming lecture): Where does the word monad come from? I know that a monad is a monoid in the category of endofunctors, but what is the logic monoid => monad? Btw, I frequently encounter monads in a categories of functors which are not endofunctors. An example are finite dimensional vectorspaces which can be constructed via a monoid in the category of functors FinSet -> Set, here I is the embedding and (x) can be constructed from the left kan extension and composition. The unit is given by the Kronecker delta and join can be constructed from Matrix multiplication. Should one call these beasts monads as well? Is there a good reference for this type of construction? Cheers, Thorsten From rrosebru@mta.ca Wed Apr 1 13:54:55 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 01 Apr 2009 13:54:55 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lp3i9-0006D2-0L for categories-list@mta.ca; Wed, 01 Apr 2009 13:54:45 -0300 MIME-version: 1.0 From: =?ISO-8859-1?Q?Jos=E9_Luiz_Fiadeiro?= To: categories@mta.ca Subject: categories: Lecturer in Computer Science, University of Leicester Date: Wed, 01 Apr 2009 17:48:39 +0100 Content-type: text/plain; charset=ISO-8859-1; format=flowed; delsp=yes Content-transfer-encoding: quoted-printable Sender: categories@mta.ca Precedence: bulk Reply-To: =?ISO-8859-1?Q?Jos=E9_Luiz_Fiadeiro?= Message-Id: Status: O X-Status: X-Keywords: X-UID: 2 Lecturer in Computer Science Department of Computer Science University of Leicester Salary Grade 8: =A335,469 to =A343,622 p.a. Available from: 1 September 2009 Ref: A4140 The successful candidate will have a strong or promising research =20 record in computer science, with a background in formal foundations =20 (either algorithms and complexity, or semantics of programming or =20 modelling languages), and will be able to contribute to undergraduate =20= and postgraduate teaching and supervision in software engineering. If you wish to apply, download an application form and further =20 information from www.le.ac.uk/personnel/jobs or contact Personnel =20 Services on recruitment3@le.ac.uk. Closing Date: Friday 1 May 2009 Times Higher Education University of the Year 2008/09 From rrosebru@mta.ca Thu Apr 2 09:10:58 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 02 Apr 2009 09:10:58 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LpLjW-00036Q-69 for categories-list@mta.ca; Thu, 02 Apr 2009 09:09:22 -0300 Date: Wed, 1 Apr 2009 13:13:55 -0500 (EST) From: Michael Barr To: Thorsten Altenkirch , Subject: categories: Re: Where does the term monad come from? MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: categories@mta.ca Precedence: bulk Reply-To: Michael Barr Message-Id: Status: RO X-Status: X-Keywords: X-UID: 3 I have told this story many times, but I guess one more can't hurt. Of course, it was originally used by Leibniz to describe the set of infintesimals surrounding an ordinary point. In the summer (or maybe late spring, the Oberwohlfach records will show this) of 1966, there was a category meeting there. It was, as far as I know, the third meeting ever devoted to categories. The first was the first Midwest Category meeting, an invitation affair that consisted of five people from Urbana (Jon Beck, John Gray, Alex Heller, Max Kelly, and me), John Isbell and Fred Linton visiting Chicago that year, and a couple people from U. Chicago, Mac Lane who was the host and arranged to pay our expenses, Dick Swan, and maybe a couple others. The second was in La Jolla and this was the third. The attendance consisted of practically everyone in the world who had any interest in categories, with the notable exception of Charles Ehresmann. What, with one exception, most categorists call monads had by that time been called "Standard constructions", "fundamental constructions" (in a little-known paper by Jean-Marie Maranda pointed out to me by Peter Huber), and, of course, "Triples". The latter was created by Eilenberg-Moore and I once asked Sammy (who was known to agonize over good terminology--e.g. "Exact") why. He answered that the concept seemed to be of little importance, so he and John Moore spent no time on it! So much for the predictive ability of a great mathematician. At any rate, the big unanswered question of the meeting, where the importance of the concept was becoming clear (Jon and I had proved our Acyclic models theorem, for example, and the fact of the triplebleness of compact Hausdorff spaces over sets, along with many mor familiar examples), the search was on for a better name. We tried many ideas (mine was "Standard Natural Algebraic Functor with Unit" (try the acronym). One day at lunch or dinner I happened to be sitting next to Jean Benabou and he turned to me and said something like "How about `monad'?" I thought about and said it sounded pretty good to me. (Yes, I did.) So Jean proposed it to the general audience and there was general agreement. It suggested "monoid" of course and it is a monoid in a functor category. The one dissenter was Jon Beck, who had invested as much into studying them as anyone. His argument was that while "triples" was not a good name, "monad" wasn't either and we shouldn't change the name from a poor one to a mediocre one, but instead continue to search for a better one. Out of solidarity with Jon (we collaborated on several papers), I continued to use "triple". SLN 80 was (and is) known as the "Zurich Triples Book". By 1980, Jon was no longer doing serious mathematics and I was ready to change. Except that the book title "Toposes, Triples and Theories" was too attactive to let go of. Try "Toposes, Monads and Theories". Incidentally, Peter May also claims to have invented the term. Treat that claim with the contempt it deserves. The most charitable explanation I have is that he heard it from Mac Lane, forgot that he had and then came up with it later. On Wed, 1 Apr 2009, Thorsten Altenkirch wrote: > A question just came up at the Midland Graduate School (actually in > the functional programming lecture): > Where does the word monad come from? > > I know that a monad is a monoid in the category of endofunctors, but > what is the logic monoid => monad? > > Btw, I frequently encounter monads in a categories of functors which > are not endofunctors. An example are finite dimensional vectorspaces > which can be constructed via a monoid in the category of functors > FinSet -> Set, here I is the embedding and (x) can be constructed from > the left kan extension and composition. > The unit is given by the Kronecker delta and join can be constructed > from Matrix multiplication. Should one call these beasts monads as > well? Is there a good reference for this type of construction? > > Cheers, > Thorsten > > From rrosebru@mta.ca Thu Apr 2 09:12:55 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 02 Apr 2009 09:12:55 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LpLmt-0003MK-3y for categories-list@mta.ca; Thu, 02 Apr 2009 09:12:51 -0300 Date: Wed, 1 Apr 2009 20:45:33 +0200 (CEST) Subject: categories: Re: Where does the term monad come from? From: Johannes.Huebschmann@math.univ-lille1.fr To: "Thorsten Altenkirch" , MIME-Version: 1.0 Content-Type: text/plain;charset=utf-8 Content-Transfer-Encoding: quoted-printable Sender: categories@mta.ca Precedence: bulk Reply-To: Johannes.Huebschmann@math.univ-lille1.fr Message-Id: Status: O X-Status: X-Keywords: X-UID: 4 >From my recollections, the terminology monad was suggested by P. May as a replacement for triple. The terminology was intended to match with "operad". At the time, S. Mac Lane has taken up that suggestion. In his book "Categories for the working mathematician" Mac Lane uses the terminology monad and comonad rather than triple and cotriple. If Peter May participates in this board I am sure he will react. Johannes > A question just came up at the Midland Graduate School (actually in > the functional programming lecture): > Where does the word monad come from? > > I know that a monad is a monoid in the category of endofunctors, but > what is the logic monoid =3D> monad? > > Btw, I frequently encounter monads in a categories of functors which > are not endofunctors. An example are finite dimensional vectorspaces > which can be constructed via a monoid in the category of functors > FinSet -> Set, here I is the embedding and (x) can be constructed from > the left kan extension and composition. > The unit is given by the Kronecker delta and join can be constructed > from Matrix multiplication. Should one call these beasts monads as > well? Is there a good reference for this type of construction? > > Cheers, > Thorsten > > > From rrosebru@mta.ca Thu Apr 2 09:13:42 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 02 Apr 2009 09:13:42 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LpLne-0003Qr-Rk for categories-list@mta.ca; Thu, 02 Apr 2009 09:13:38 -0300 Date: Wed, 01 Apr 2009 20:47:12 +0100 From: Venanzio Capretta MIME-Version: 1.0 To: Thorsten Altenkirch , Subject: categories: Re: Where does the term monad come from? Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: Venanzio Capretta Message-Id: Status: O X-Status: X-Keywords: X-UID: 5 The philosopher Gottfried Leibniz believed that every entity in the Universe is a separate substance that doesn't interact with others. He called these substances "monads". All properties and events that happen to a monad are implicit in its nature from its creation. So if an apple falls from a tree and bounces off my head, there is actually no contact: the apple-monad bounces by itself without the help of my head and the Venanzio-monad feels pain without the intervention of the apple. All monads are synchronized from creation by the wisdom of God. This implies that every monad has an internal representation of every entity in the universe and these representations can never influence objects outside the monad. The analogy with our monads should be evident! Thorsten Altenkirch wrote: > A question just came up at the Midland Graduate School (actually in > the functional programming lecture): > Where does the word monad come from? > > I know that a monad is a monoid in the category of endofunctors, but > what is the logic monoid => monad? > > Btw, I frequently encounter monads in a categories of functors which > are not endofunctors. An example are finite dimensional vectorspaces > which can be constructed via a monoid in the category of functors > FinSet -> Set, here I is the embedding and (x) can be constructed from > the left kan extension and composition. > The unit is given by the Kronecker delta and join can be constructed > from Matrix multiplication. Should one call these beasts monads as > well? Is there a good reference for this type of construction? > > Cheers, > Thorsten > > From rrosebru@mta.ca Thu Apr 2 09:14:50 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 02 Apr 2009 09:14:50 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LpLoj-0003Wj-Ju for categories-list@mta.ca; Thu, 02 Apr 2009 09:14:45 -0300 Date: Wed, 01 Apr 2009 23:19:50 +0200 From: burroni@math.jussieu.fr To: Thorsten Altenkirch , Subject: categories: Re: Where does the term monad come from? 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: categories@mta.ca Precedence: bulk Reply-To: burroni@math.jussieu.fr Message-Id: Status: O X-Status: X-Keywords: X-UID: 6 Cher Thorsten, toutes mes excuses pour ce message en fran=E7ais. Le terme "monade" a =E9t=E9 employ=E9 par Benabou (LNM Springer no 47, si je= =20 ne me trompe) et dans un sens abstrait : pseudofoncteur 1 --> B de la =20 bicat=E9gorie finale 1 vers une bicat=E9gorie arbitraire B. Par la suite =20 il a =E9t=E9 convenu de le r=E9sever au cas particulier o=F9 B=3DCat (en =20 remplacement du terme "triple"). A mon avis, le terme est remarquable car il combine ceux de "monoides" =20 et de "monades", concept utilis=E9 par Leibnitz, mais qui, ind=E9pendement = =20 de l'usage fait par ce philosophe, signifie : unit=E9 simple, =20 ind=E9composable. Cette simplicit=E9, cette ind=E9composabilit=E9 est celle = de =20 la bicat=E9gorie 1. Aujourd'hui, on appelle monoide, les monades au sens g=E9n=E9ral de =20 Benabou. (Personnellement, je ne trouve cela imparfait car un vrai =20 monoide est une structure beaucoup plus riche : exemple x |--> x^2 n'a =20 pas de sens en general.) amiti=E9s, Albert Thorsten Altenkirch a =E9crit=A0: > A question just came up at the Midland Graduate School (actually in > the functional programming lecture): > Where does the word monad come from? > > I know that a monad is a monoid in the category of endofunctors, but > what is the logic monoid =3D> monad? > > Btw, I frequently encounter monads in a categories of functors which > are not endofunctors. An example are finite dimensional vectorspaces > which can be constructed via a monoid in the category of functors > FinSet -> Set, here I is the embedding and (x) can be constructed from > the left kan extension and composition. > The unit is given by the Kronecker delta and join can be constructed > from Matrix multiplication. Should one call these beasts monads as > well? Is there a good reference for this type of construction? > > Cheers, > Thorsten From rrosebru@mta.ca Fri Apr 3 10:00:00 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 03 Apr 2009 10:00:00 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lpixb-00051P-BA for categories-list@mta.ca; Fri, 03 Apr 2009 09:57:27 -0300 Date: Thu, 2 Apr 2009 07:51:22 -0500 (CDT) From: Peter May To: categories@mta.ca Subject: categories: Monads Sender: categories@mta.ca Precedence: bulk Reply-To: Peter May Message-Id: Status: RO X-Status: X-Keywords: X-UID: 7 Michael, where on earth did that piece of contemptible writing come from. I never claimed to invent the term monad. I did invent the term operad, as a portmanteau of operation and monad. And I convinced MacLane to change from the silly term `triple' to `monad' in Categories for the working mathematician. He is not here to corroborate, but look at his note on terminology, page 138 of the second edition: ``The frequent but unfortunate use of the word `triple' in this sense has achieved a maximum of needless confusion, what with the conflict with ordered triple, plus the use of associated terms such as ``triple derived functor''for functors which are not three times derived from anything in the world. Hence the term monad.'' Michael, shame on you! Peter May From rrosebru@mta.ca Fri Apr 3 10:00:01 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 03 Apr 2009 10:00:01 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lpiyy-00058H-1M for categories-list@mta.ca; Fri, 03 Apr 2009 09:58:52 -0300 Date: Thu, 02 Apr 2009 09:31:00 -0400 From: jim stasheff MIME-Version: 1.0 To: Johannes.Huebschmann@math.univ-lille1.fr, Thorsten Altenkirch , categories@mta.ca Subject: categories: Re: Where does the term monad come from? Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: jim stasheff Message-Id: Status: O X-Status: X-Keywords: X-UID: 8 Whereas my recollection (from those dear dim days beyond recall when I was present on a weekly basis for ND about that time) was that the terminology went from Mac Lane to May with operad to match monad as I recall, Mac Lane liked monad because of the philosophical connection Leibniz as philosopher not as mathematician? * Monad (Greek philosophy) a term used by ancient philosophers Pythagoras, Parmenides, Xenophanes, Plato, Aristotle, and Plotinus as a term for God or the first being, or the totality of all being. * Monism, the concept of "one essence" in the metaphysical and theological theory * Monad (Gnosticism), the most primal aspect of God in Gnosticism ****** Monadology, a book of philosophy by Gottfried Leibniz in which monads are a basic unit of perceptual reality * Monadologia Physica by Immanuel Kant * The Cup or Monad, a text in the Corpus Hermetica from the Wiki Johannes.Huebschmann@math.univ-lille1.fr wrote: > >From my recollections, the terminology monad was suggested by P. May > as a replacement for triple. > The terminology was intended to match with "operad". > At the time, S. Mac Lane has taken up that suggestion. > In his book "Categories for the working mathematician" > Mac Lane uses the terminology monad and comonad rather than triple > and cotriple. > > If Peter May participates in this board I am sure he will react. > > Johannes > >> A question just came up at the Midland Graduate School (actually in >> the functional programming lecture): >> Where does the word monad come from? >> >> I know that a monad is a monoid in the category of endofunctors, but >> what is the logic monoid => monad? >> >> Btw, I frequently encounter monads in a categories of functors which >> are not endofunctors. An example are finite dimensional vectorspaces >> which can be constructed via a monoid in the category of functors >> FinSet -> Set, here I is the embedding and (x) can be constructed from >> the left kan extension and composition. >> The unit is given by the Kronecker delta and join can be constructed >> from Matrix multiplication. Should one call these beasts monads as >> well? Is there a good reference for this type of construction? >> >> Cheers, >> Thorsten >> >> >> >> > > > > > From rrosebru@mta.ca Fri Apr 3 10:00:20 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 03 Apr 2009 10:00:20 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lpj0K-0005FF-PE for categories-list@mta.ca; Fri, 03 Apr 2009 10:00:16 -0300 Date: Fri, 03 Apr 2009 15:28:26 +1100 Subject: categories: Re: Where does the term monad come from? From: Steve Lack To: Thorsten Altenkirch , Mime-version: 1.0 Content-type: text/plain; charset="US-ASCII" Content-transfer-encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: Steve Lack Message-Id: Status: O X-Status: X-Keywords: X-UID: 9 Dear All, As usual, there have been plenty of people with comments about history. There was also a second part to the question: > > Btw, I frequently encounter monads in a categories of functors which > are not endofunctors. An example are finite dimensional vectorspaces > which can be constructed via a monoid in the category of functors > FinSet -> Set, here I is the embedding and (x) can be constructed from > the left kan extension and composition. > The unit is given by the Kronecker delta and join can be constructed > from Matrix multiplication. Should one call these beasts monads as > well? Is there a good reference for this type of construction? The category of functors from FinSet to Set is equivalent to the category of endofunctors of Set which preserve filtered colimits: such endofunctors are usually called finitary. Thus a monoid in [FinSet,Set] with respect to this tensor product is the same thing as a monad on Set whose endofunctor part is finitary: this is called a finitary monad. These finitary monads on Set are equivalent to Lawvere theories and so in turn to (finitary, single-sorted) varieties. Finitary monads can also be considered on other base categories than Set, especially on locally finitely presentable ones. It is true that vector spaces are the algebras for a finitary monad on Set. There is no need to restrict to finite-dimensional vector spaces; in fact it is not true that there is a monad on Set whose algebras are the finite-dimensional vector spaces. Steve. From rrosebru@mta.ca Fri Apr 3 10:01:20 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 03 Apr 2009 10:01:20 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lpj1I-0005KM-Bx for categories-list@mta.ca; Fri, 03 Apr 2009 10:01:16 -0300 Date: Fri, 03 Apr 2009 15:33:13 +1100 Subject: categories: Re: Where does the term monad come from? From: Steve Lack To: , Thorsten Altenkirch , Mime-version: 1.0 Content-type: text/plain; charset="ISO-8859-1" Content-transfer-encoding: quoted-printable Sender: categories@mta.ca Precedence: bulk Reply-To: Steve Lack Message-Id: Status: O X-Status: X-Keywords: X-UID: 10 Dear All, Just another quick comment about monads: On 2/04/09 8:19 AM, "burroni@math.jussieu.fr" wrote: > Cher Thorsten, >=20 > toutes mes excuses pour ce message en fran=E7ais. >=20 > Le terme "monade" a =E9t=E9 employ=E9 par Benabou (LNM Springer no 47, si je > ne me trompe) et dans un sens abstrait : pseudofoncteur 1 --> B de la > bicat=E9gorie finale 1 vers une bicat=E9gorie arbitraire B. Par la suite > il a =E9t=E9 convenu de le r=E9sever au cas particulier o=F9 B=3DCat (en > remplacement du terme "triple"). Some people may reserve monad for the case B=3DCat, but not all. After Benabo= u demonstrated the incredible importance of this idea in various B, the theor= y of monads in 2-categories/bicategories has been widely developed, starting (I believe) with Ross Street's "Formal theory of monads". Steve. From rrosebru@mta.ca Sat Apr 4 11:57:27 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 04 Apr 2009 11:57:27 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lq7GV-00001g-7i for categories-list@mta.ca; Sat, 04 Apr 2009 11:54:35 -0300 Mime-Version: 1.0 (Apple Message framework v753.1) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit From: Ross Street Subject: categories: RE: Monads Date: Sat, 4 Apr 2009 14:31:39 +1100 To: categories Sender: categories@mta.ca Precedence: bulk Reply-To: Ross Street Message-Id: Status: RO X-Status: X-Keywords: X-UID: 11 I hope I can add some jigsaw pieces towards the history of the term "monad" in category theory without offending anyone. 1) It is clearly a fact that the term "monad" is used in Benabou's paper SLNM 47 (1967). He recognized that it is a morphism of bicategories from the terminal category 1. 2) I have a clear memory that Mac Lane told me (perhaps at Chicago while I was a postdoc at Champaign-Urbana 1968-69) that Benabou courteously asked him (possibly by airmail, maybe by phone call, maybe at a conference) whether Mac Lane would mind whether he used the term "bicategory" in the sense we now use it. Mac Lane had used "bicategory" to mean a category with two distinguished classes of morphisms: roughly speaking, what we now call a category with a factorization system. Mac Lane told Benabou he did not mind. So Benabou used it in SLNM 47. 3) Less clearly I remember Mac Lane said Benabou also suggested the term "monad" for use in SLNM 47. 4) It is again my clear memory that, in his lecture marathon at the Summer School on Category Theory at Bowdoin College (Maine, mid-1969), Mac Lane expressed strong dislike for the term "triple" but had not really settled on a term. Mac Lane actually used the term "triad" in his lectures at Bowdoin. 5) At CT08, Lawvere told me Eilenberg suggested the term "monad". Best wishes, Ross From rrosebru@mta.ca Sun Apr 5 18:01:25 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 05 Apr 2009 18:01:25 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LqZQG-0005H4-MP for categories-list@mta.ca; Sun, 05 Apr 2009 17:58:32 -0300 Date: Sat, 04 Apr 2009 11:45:47 -0400 From: jim stasheff MIME-Version: 1.0 To: Ross Street , Subject: categories: RE: Monads Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: jim stasheff Message-Id: Status: O X-Status: X-Keywords: X-UID: 12 Ross Street wrote: > > I hope I can add some jigsaw pieces towards the history of the term > "monad" in category theory without offending anyone. > > ... > Best wishes, > Ross > > Some version of this and other responses whould be added to the Wiki I'm not up to the job. jim From rrosebru@mta.ca Mon Apr 6 13:38:38 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 06 Apr 2009 13:38:38 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lqrm9-0006Rm-Gj for categories-list@mta.ca; Mon, 06 Apr 2009 13:34:21 -0300 Date: Mon, 6 Apr 2009 06:52:25 +0200 (MEST) From: Patrik Eklund To: categories@mta.ca Subject: categories: Re: Where does the term monad come from? MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: categories@mta.ca Precedence: bulk Reply-To: Patrik Eklund Message-Id: Status: O X-Status: X-Keywords: X-UID: 13 "Operads" are like sets of operations. A monad is an extension of a functor. If the functor is the term functor, then the operations of the signature lies inside the functor, and the "operations" eta and mu are identities, or at least something very isomorphic to identities. In the filter functor eta is point filters and mu is Kowalsky's diagonalization. In my view there is no logic monoid => monad, and I cannot see the full idea behind using "operads", so help me Mona. Patrik On Thu, 2 Apr 2009, jim stasheff wrote: > Whereas my recollection (from those dear dim days beyond recall when I > was present on a weekly basis for ND about that time) > was that the terminology went from Mac Lane to May with > operad to match monad > > as I recall, Mac Lane liked monad because of the philosophical connection > Leibniz as philosopher not as mathematician? > > * Monad (Greek philosophy) a term used by ancient philosophers > Pythagoras, Parmenides, Xenophanes, Plato, Aristotle, and Plotinus as a > term for God or the first being, or the totality of all being. > * Monism, the concept of "one essence" in the metaphysical and > theological theory > * Monad (Gnosticism), the most primal aspect of God in Gnosticism > ****** Monadology, a book of philosophy by Gottfried Leibniz in > which monads are a basic unit of perceptual reality > * Monadologia Physica by Immanuel Kant > * The Cup or Monad, a text in the Corpus Hermetica > from the Wiki > > From rrosebru@mta.ca Mon Apr 6 13:44:16 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 06 Apr 2009 13:44:16 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LqrvV-0007gK-Bb for categories-list@mta.ca; Mon, 06 Apr 2009 13:44:01 -0300 Date: Mon, 6 Apr 2009 06:33:17 +0200 (MEST) From: Patrik Eklund To: categories@mta.ca Subject: categories: RE: Monads MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=iso-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable Sender: categories@mta.ca Precedence: bulk Reply-To: Patrik Eklund Message-Id: Status: O X-Status: X-Keywords: X-UID: 14 [Note from moderator: this thread has strayed; although this post is allowed, comments closer to categories (not Kant's) are preferred.] Reference to Leibniz is nice, and so is going back even more in history.=20 Going forward into modern history leads to problems of who actually cause= d=20 what. Probably because we then tend to mix history and politics. Anyway, also having googled, I found this about Leibniz: =A7. 1. Die Monaden (Das Worte Monade oder Monas) wovon wir allhier reden= =20 werden / sind nichts anders als einfache Substanzen / woraus die zusammen= =20 gesetzten Dinge oder composita bestehen. Unter dem Wort / einfach /=20 verstehet man dasjenige / welches keine Teile hat. "sind nichts anders als einfache Substanzen" "is nothing but simple substances" They are, but it is not a mathematical statement. "woraus die zusammen gesetzten Dinge oder composita bestehen" "using which you put them together or compose(!) them together" Now he is cooking. Monad compositions are important. Leibniz and Beck=20 working together, I like it. This is closer to mathematics. "verstehet man dasjenige / welches keine Teile hat" "is to be understood as something which doesn't have subparts" I am sure there are non-trivial monads which are not composed (in Beck's=20 sense) by other non-trivial monads. But more interestingely, composed=20 monads are indeed monads, and even worse (from leibniz point of=20 view) submonads do exist, like the filter monad being submonad to the=20 ultrafilter monad (with the astonishing fact, yes, I know, I am repreatin= g=20 myself, that their respective algebras are Scott lattices and compact=20 Hausdorff spaces). So, basically I like Leibniz, even if he was wrong at this point. History= =20 is not easy. We say "Rome was destroyed" and we frequently say by the=20 goths. Saying that leads us to ask "how could it be destroyed". Seldom do= =20 we hear "how could it stay alive so long". Best, Patrik PS "Monas" seems mostly to be used for a sailing boat, the "Kiel", and=20 "the Mona" is Louvre in Paris. From rrosebru@mta.ca Fri Apr 3 20:46:01 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 03 Apr 2009 20:46:01 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lpt4i-00078p-EC for categories-list@mta.ca; Fri, 03 Apr 2009 20:45:28 -0300 From: Marta Bunge To: , Subject: categories: RE: Monads Date: Fri, 3 Apr 2009 09:56:46 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Sender: categories@mta.ca Precedence: bulk Reply-To: Marta Bunge Message-Id: Status: O X-Status: X-Keywords: X-UID: 15 Dear all=2CSomething to corroborate MacLane's abhorrence of the word "tripl= e" is=2C in my view=2C his refusal to communicate my first paper (Marta Bun= ge=2C Relative Functor Categories and Categories of Algebras=2C J.of Algebr= a 11 (1969) 64-101) unless I changed the word "triple" in it for that of "m= onad". In order to show my independence (!)=2C yet wishing to have the pape= r published=2C I changed "triple" back to "standard construction". This he = accepted without objections. Nowadays I use monads like everybody else. I h= ave no idea which=2C among the many possible reasons suggested in categorie= s=2C was MacLane's reason for insisting on "monads"=2C whether philosophica= l or mathematical. However=2C his acceptance of my use of "standard constru= ction" suggests that his dislike of "triple" was stronger than his preferen= ce for "monad". Cordial regards=2CMarta Bunge From rrosebru@mta.ca Fri Apr 3 20:46:01 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 03 Apr 2009 20:46:01 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lpt3Z-00075E-71 for categories-list@mta.ca; Fri, 03 Apr 2009 20:44:17 -0300 Date: Fri, 03 Apr 2009 15:55:16 +0200 From: burroni@math.jussieu.fr To: jim stasheff , categories@mta.ca Subject: categories: Re: Where does the term monad come from? MIME-Version: 1.0 Content-Type: text/plain;charset=UTF-8; DelSp="Yes"; format="flowed" Content-Disposition: inline Content-Transfer-Encoding: quoted-printable Sender: categories@mta.ca Precedence: bulk Reply-To: burroni@math.jussieu.fr Message-Id: Status: RO X-Status: X-Keywords: X-UID: 16 > * The Cup or Monad, a text in the Corpus Hermetica > from the Wiki Je trouverais normal que le mod=C3=A9rateur ne permette pas cette =20 intervention. C'est en partie =C3=A0 titre de plaisanterie et en compl=C3=A9= ment =20 =C3=A0 une information de Jim que je la fait. Pour des sources mystiques sur la monade : http://www.esotericarchives.com/dee/monade.htm on y trouve des th=C3=A9or=C3=A8mes in=C3=A9dits en th=C3=A9orie des cat=C3= =A9gories. Albert From rrosebru@mta.ca Fri Apr 3 20:46:54 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 03 Apr 2009 20:46:54 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lpt5u-0007CZ-LP for categories-list@mta.ca; Fri, 03 Apr 2009 20:46:42 -0300 Date: Fri, 03 Apr 2009 10:07:56 -0400 From: jim stasheff MIME-Version: 1.0 To: Peter May , categories@mta.ca, Subject: categories: Re: Monads Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: jim stasheff Message-Id: Status: RO X-Status: X-Keywords: X-UID: 17 Peter May wrote: > I never claimed to invent the term monad. I did > invent the term operad, as a portmanteau of operation and monad. > Peter May > > which I am happy to confirm I think I was there at the time or at least nearby jim Rainer, Is this covered in your memoir of those miravulaous years? jim From rrosebru@mta.ca Fri Apr 3 20:47:50 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 03 Apr 2009 20:47:50 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lpt6r-0007GN-7W for categories-list@mta.ca; Fri, 03 Apr 2009 20:47:41 -0300 From: Michael Barr To: Peter May , categories@mta.ca Subject: categories: Re: Monads MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: categories@mta.ca Precedence: bulk Reply-To: Michael Barr Message-Id: Date: Fri, 03 Apr 2009 20:47:41 -0300 Status: RO X-Status: X-Keywords: X-UID: 18 Sorry if this offended you, but I heard from several places that you claimed the invention of the term. You will note that one other respondent credited it to you, so there must have been a meme to that effect. If you never made that claim then I truly apologize. Perhaps people confused "monad" with "operad", which I do believe you invented. Michael From rrosebru@mta.ca Mon Apr 6 21:23:28 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 06 Apr 2009 21:23:28 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lqz3Z-0002Pn-TB for categories-list@mta.ca; Mon, 06 Apr 2009 21:20:49 -0300 MIME-Version: 1.0 Date: Mon, 6 Apr 2009 13:24:10 -0700 Subject: categories: Re: Where does the term monad come from? From: John Baez To: categories@mta.ca Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: John Baez Message-Id: Status: O X-Status: X-Keywords: X-UID: 19 Patrik Eklund wrote: In my view there is no logic monoid => monad... It's pretty much been said, but I'll say it again: We can generalize the concept of monoid from Set to any monoidal category and then to any bicategory. A monoid in Cat is then a monad. Indeed, most people seem to call a "monoid" in a bicategory a "monad". Best, jb From rrosebru@mta.ca Tue Apr 7 08:25:21 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 07 Apr 2009 08:25:21 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lr9Ol-0001OC-46 for categories-list@mta.ca; Tue, 07 Apr 2009 08:23:23 -0300 Date: Mon, 6 Apr 2009 23:06:37 -0300 (ADT) From: RJ Wood To: categories@mta.ca cc: Rj Wood Subject: categories: Re: Where does the term monad come from? MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: categories@mta.ca Precedence: bulk Reply-To: RJ Wood Message-Id: Status: RO X-Status: X-Keywords: X-UID: 20 John Baez wrote: It's pretty much been said, but I'll say it again: We can generalize the concept of monoid from Set to any monoidal category and then to any bicategory. A monoid in Cat is then a monad. Indeed, most people seem to call a "monoid" in a bicategory a "monad". Best, jb John, given the didactic nature of this thread, I think we should be more precise about what you mean by `a "monoid" in a bicategory'. For a bicategory B and an object X therein, B(X,X) (together with composition, 1_X, and the inherited constraints of B) i s a monoidal category and a monad in B is an object X in B together with a monoid in B(X,X). Rj From rrosebru@mta.ca Tue Apr 7 08:26:11 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 07 Apr 2009 08:26:11 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lr9Qq-0001Ub-DZ for categories-list@mta.ca; Tue, 07 Apr 2009 08:25:32 -0300 Date: Tue, 07 Apr 2009 00:32:17 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Where does the term monad come from? Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: Vaughan Pratt Message-Id: Status: O X-Status: X-Keywords: X-UID: 21 Patrik Eklund wrote: > "Operads" are like sets of operations. > > A monad is an extension of a functor. If the functor is the term functor, > then the operations of the signature lies inside the functor, and the > "operations" eta and mu are identities, or at least something very > isomorphic to identities. > > In the filter functor eta is point filters and mu is Kowalsky's > diagonalization. > > In my view there is no logic monoid => monad, and I cannot see the > full idea behind using "operads", so help me Mona. In one paragraph, a monad can be understood as a set T of operations graded by their arity X, that is, a variable set T(X) where the set X is the parameter of variation. This set is made a monoid by the operation of substitution interpreted as the multiplication of the monoid. Substitution is associative (terms of height three can be built top-down or bottom-up) and has a two-sided identity interpretable ambiguously as the identity operation of T (when applied to the top of a term) and substitution of variables for themselves (when applied to the bottom of a term). The back-to-back talks of Walter Taylor on his general theory of varieties and Fred Linton on monads at the Universal Algebra and Category Theory (UACT) meeting held at MSRI in 1993 were like ships passing in the night. No one thought to mention, either before, during, or after the talks, that they were describing essentially the same thing (or if they did both George McNulty and I missed it), with the result that many of the algebraists at the meeting just assumed that these were unrelated talks. Monads can be explained in terms of their associated Kleisli category, or their Eilenberg-Moore category, or as the composition UF of any set-valued functor U with its left adjoint. After Fred's talk I had lunch with George and tried out the third of these on him. However we got bogged down in the definition of adjunction. In hindsight I think the quickest way to explain a monad to an algebraist is to do so directly in terms of T, \mu, and \eta, without the distraction of the additional machinery of the three above methods. It would go something like the following, which of the above three is closest to the Kleisli category approach. I'll ignore the inconsistent monads, those axiomatized by x=y, for which T(X) = 1 for nonempty X and T(0) <= 1. A monad specifies the language and equations of an equational theory. The functor T specifies the language by providing for each set X the set of operations (more properly polynomials or abstract terms) of arity X, e.g. T(2) is the set of binary operations of the theory. The multiplication \mu_X: T(T(X)) --> T(X) specifies the theory by mapping terms of height at most two to operations (identified with terms of height at most one). Terms s and t of height two identified by \mu, e.g. x(y+z) = xy+xz in the case of ring theory, constitute the axioms s = t of the theory determined by \mu. Hardware types and visual thinkers can picture T(X) as a black box containing all operations of arity X. X can be thought of as a row (or any other layout, I like the unit interval [0,1] of reals for picturing an uncountable set as a row) of input sockets on one side and T(X) as a row of output plugs on the opposite side, one per operation. The unit \eta_X: X --> T(X) at X, necessarily an injection, ensures that the operations include the variables, qua (formal) projections. Thinking of T(X) as consisting of terms, define the height of each variable to be zero and that of the remaining operations of T(X) as one. Since T can take any set of variables it can take in particular T(X), whence there is also a box with input set T(X) and output set T(T(X)). The boxes containing T(X) and T(T(X)) can be plugged together to form a single black box with set X of inputs and set T(T(X)) of outputs. However this latter set must now be interpreted as consisting of entities of arity X instead of arity T(X). Viewed syntactically (taking into account the separate contents of the two boxes and how they attach) we can consider the outputs of T(T(X)) as terms of height two in the variables in X, or rather at most two since the unit of the monad embeds X in T(X) and T(X) in T(T(X)). One can then ask whether any of these terms realize some operation not among those of T(X). The function \mu_X: T(T(X)) --> T(X) accomplishes three things. (i) It interprets every term of height up to two as an operation of T(X), a form of abstract evaluation that hides the two-level term structure. (ii) In so doing it answers the above question in the negative: no new operations, all terms of height up to two realize operations already present in T(X). In this sense T as a graded set of operations is closed under substitution. (iii) As noted above it axiomatizes the equational theory associated with the monad with all equations of that theory involving terms of height at most two, namely all equations s = t such that \mu_X maps s and t to the same operation of T(X). \mu can be extended to evaluate terms of height h inductively. If \mu_h: T^h --> T evaluates terms of height up to h then the vertical composite \mu T(\mu_h): T^{h+1} --> T^2 --> T evaluates terms of height up to h+1, starting from \mu_2 = \mu. (So \mu_h = \mu T(\mu) T^2(\mu)...T^{h-2}(\mu).) This is the categorical counterpart of using equational logic to inductively build up the height of equations in the theory one level at a time via \mu, starting from the equations constituting the kernel of \mu. (Although height is always finite there is no such restriction on arity and hence on width of a term, which can be any set. Even for a finitary monad an operation can take uncountably many arguments, e.g. for the monad for Vct_R as the variety of vector spaces over the reals, each operation in T(T(2)) takes as many parameters as there are linear combinations ax+by, namely uncountably many, though it depends on only finitely many of them because Vct_R is a finitary variety in the sense Steve Lack referred to on Friday.) If the boxes really do consist of operations then the two pluggings required to form a chain T(X), T(T(X)), T(T(T(X))) of three black boxes should have the same operational effect regardless of the order in which they're performed. The associativity axiom for a monad enforces this property of black boxes containing operations. The above inductive definition of \mu_h was bottom-up (T^{h+1} = TT^h) but there is an equivalent top-down one (T^h T) producing the same \mu_h. The Eilenberg-Moore category of a monad is the variety of algebras it axiomatizes, modulo details of the treatment of the associated signature. In general the signature will be a proper class but in many cases encountered in practice one can pick out of this class a (small) set sufficient for a basis of operations, e.g. +, -, and the scalar multiplications for Vct_k, or NAND and a constant for Boolean algebra. The variety of sigma-algebras is not finitary although it can still be furnished with a small signature, but this is not possible for the varieties of complete semilattices and of complete atomic Boolean algebras, which old-school algebraists would not consider varieties for that reason. The Kleisli category is the full subcategory of the variety consisting of its free algebras. The latter is intimately linked to the above intuitions, and amounts to an equivalent way of seeing that T is closed under substitution by formulating substitution as the composition of the Kleisli category. The essential point of departure from the usual notion of a monoid as a fixed set is that for the type T^2 --> T of multiplication, T^2 is defined via composition instead of cartesian product. Monads are therefore simply monoids adapted in this way to accommodate their variability. Monads need not be sets. Just as a ring can be defined as a monoid object in Ab, with the domain of multiplication being formed via cartesian product in Ab, so can a monad be a monoid object in the category of endofunctors of Ab, with the domain of multiplication being formed by functor composition in Ab^Ab. Although not all categories have finite products, T^2 is defined for every category C and endofunctor T: C --> C. Vaughan Pratt From rrosebru@mta.ca Tue Apr 7 08:27:05 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 07 Apr 2009 08:27:05 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lr9Ri-0001Xl-KV for categories-list@mta.ca; Tue, 07 Apr 2009 08:26:26 -0300 Date: Tue, 07 Apr 2009 08:50:29 +0100 From: Tim Porter To: "categories@mta.ca" Subject: categories: L'Aquila category theorists? MIME-Version: 1.0 Content-Type: text/plain;charset=ISO-8859-1;DelSp="Yes";format="flowed" Content-Disposition: inline Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: Tim Porter Message-Id: Status: O X-Status: X-Keywords: X-UID: 22 I like many other categorists have very fond memories of the welcome given us on visits to l'Aquila. Can I request that as soon as news comes through, of either sort, about our friends there, that the news be shared with this list. Tim Porter From rrosebru@mta.ca Tue Apr 7 19:53:05 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 07 Apr 2009 19:53:05 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LrK8e-0002Lf-9h for categories-list@mta.ca; Tue, 07 Apr 2009 19:51:28 -0300 Date: Tue, 07 Apr 2009 12:40:53 +0100 From: Maria Manuel Clementino MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: L'Aquila category theorists? Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: Maria Manuel Clementino Message-Id: Status: RO X-Status: X-Keywords: X-UID: 24 [Note from moderator: several posts with essentially the same message follow; the details mentioned vary slightly so all will be forwarded. Thanks to all who responded. It is a relief to know that the news is not worse.] Tim Porter wrote: > I like many other categorists have very fond memories of the welcome > given us on visits to l'Aquila. Can I request that as soon as news > comes through, of either sort, about our friends there, that the news > be shared with this list. > > Tim Porter I received a message from Anna Tozzi saying that Eraldo Giuli and her were fine. Maria Manuel Clementino From rrosebru@mta.ca Tue Apr 7 19:55:57 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 07 Apr 2009 19:55:57 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LrKCu-0002e7-Jb for categories-list@mta.ca; Tue, 07 Apr 2009 19:55:52 -0300 Mime-Version: 1.0 (Apple Message framework v753.1) To: categories@mta.ca From: luciano stramaccia Subject: categories: Re: L'Aquila category theorists? Date: Tue, 7 Apr 2009 14:27:30 +0200 Content-Transfer-Encoding: 7bit Content-Type: text/plain;charset=US-ASCII;delsp=yes;format=flowed Sender: categories@mta.ca Precedence: bulk Reply-To: luciano stramaccia Message-Id: Status: RO X-Status: X-Keywords: X-UID: 25 Dear Tim, I heard from Eraldo Giuli and Anna Tozzi yesterday morning soon after the quake. They all are safe (and their families). My be Eraldo's hause was slightly damaged, not Anna's. Anyway, they plan to move to their hauses along the coast where the quake had no effect. Luciano Il giorno 07/apr/09, alle ore 09:50, Tim Porter ha scritto: I like many other categorists have very fond memories of the welcome given us on visits to l'Aquila. Can I request that as soon as news comes through, of either sort, about our friends there, that the news be shared with this list. Tim Porter From rrosebru@mta.ca Tue Apr 7 19:56:44 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 07 Apr 2009 19:56:44 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LrKDh-0002hf-15 for categories-list@mta.ca; Tue, 07 Apr 2009 19:56:41 -0300 Date: Tue, 07 Apr 2009 09:57:32 -0400 From: Walter Tholen MIME-Version: 1.0 To: Tim Porter , categories@mta.ca Subject: categories: Re: L'Aquila category theorists? Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: Walter Tholen Message-Id: Status: O X-Status: X-Keywords: X-UID: 26 Tim - I have been in telephone contact with Eraldo Giuli. His and Anna Tozzi's families are fine. They are currently staying with their families away from the town of L'Aquila, in order to avoid the aftershocks. They do, however, report considerable damage to some of their properties in L'Aquila. Regards, Walter. Tim Porter wrote: > I like many other categorists have very fond memories of the welcome > given us on visits to l'Aquila. Can I request that as soon as news > comes through, of either sort, about our friends there, that the news > be shared with this list. > > Tim Porter From rrosebru@mta.ca Tue Apr 7 19:57:54 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 07 Apr 2009 19:57:54 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LrKEl-0002mo-12 for categories-list@mta.ca; Tue, 07 Apr 2009 19:57:47 -0300 Date: Tue, 07 Apr 2009 18:33:38 +0200 To: t.porter@bangor.ac.uk From: aurelio carboni Subject: categories: Re: L'Aquila category theorists? Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed Sender: categories@mta.ca Precedence: bulk Reply-To: aurelio carboni Message-Id: Status: O X-Status: X-Keywords: X-UID: 27 Dear Tim, I just succeeded in getting in touch by phone with Eraldo, after trying for quite a long time. Eraldo, his family and all the other colleagues are fine. Eraldo lost a family flat in the center of the town, but nobody was injured. He will not be able to be contacted by e-mail for quite a long time, and is now staying in his house at the beach nearby L'Aquila. Eraldo asked me to inform the colleagues and friends. Best wishes, Aurelio. >At 09.50 07/04/2009, you wrote: >>I like many other categorists have very fond memories of the welcome >>given us on visits to l'Aquila. Can I request that as soon as news >>comes through, of either sort, about our friends there, that the news >>be shared with this list. >> >>Tim Porter >> From rrosebru@mta.ca Tue Apr 7 20:00:29 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 07 Apr 2009 20:00:29 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LrKHI-0002yx-GH for categories-list@mta.ca; Tue, 07 Apr 2009 20:00:24 -0300 Date: Tue, 07 Apr 2009 11:10:41 -0400 From: jim stasheff MIME-Version: 1.0 To: Vaughan Pratt , categories@mta.ca Subject: categories: Re: Where does the term monad come from? Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: jim stasheff Message-Id: Status: O X-Status: X-Keywords: X-UID: 28 Vaughan Pratt wrote: > Patrik Eklund wrote: >> "Operads" are like sets of operations. >> >> A monad is an extension of a functor. If the functor is the term >> functor, ... > endofunctor T: C --> C. > > Vaughan Pratt > Someone should up date the Wiki jim From rrosebru@mta.ca Tue Apr 7 20:01:13 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 07 Apr 2009 20:01:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LrKI2-00033Q-G1 for categories-list@mta.ca; Tue, 07 Apr 2009 20:01:10 -0300 Date: Tue, 07 Apr 2009 11:44:24 -0400 From: jim stasheff User-Agent: Thunderbird 2.0.0.21 (Macintosh/20090302) MIME-Version: 1.0 To: Categories list Subject: categories: virtual special session Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: jim stasheff Message-Id: Status: O X-Status: X-Keywords: X-UID: 29 Last weekend there was a special session on Homotopical Algebra with Applications to Mathematical Physics at eh AMS meeting in Raleigh. I was unable to attend in person but Tom Lada set up via Skype and opportunity for me to participate. I strongly encourage this technology for virtual attendance at such meetings or informal equivalents. Slides from the session are becoming available at http://www4.ncsu.edu/~lada/NCSU%20special%20session.htm jim From rrosebru@mta.ca Tue Apr 7 20:02:13 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 07 Apr 2009 20:02:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LrKIv-00038G-NX for categories-list@mta.ca; Tue, 07 Apr 2009 20:02:05 -0300 Date: Tue, 7 Apr 2009 18:05:22 +0200 (CEST) Subject: categories: a workshop on commutativity of diagrams, Toulouse From: soloviev@irit.fr To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain;charset=utf-8 Content-Transfer-Encoding: quoted-printable Sender: categories@mta.ca Precedence: bulk Reply-To: soloviev@irit.fr Message-Id: Status: RO X-Status: X-Keywords: X-UID: 30 CAM-CAD Workshop on Computer Algebra Methods and Commutativity of Algebraic Diag= rams IRIT, Toulouse (France) October 2009 *** FIRST CALL FOR PARTICIPATION *** *** AND *** *** PAPERS *** Categorical diagrams have multiple applications in mathematics (algebra, topology) and computer science (models and metamodels, rewriting systems, higher order languages). Diagrams (understood less strictly) can be found in physics, chemistry and other scientific domains. One meets many similar problems in computer-assisted treatment of diagrams in all these domains concerning algorithms, graphic interfaces, interaction with systems of computer algebra and other software. In spite of importance of diagrammatic methods, they are relatively little developped and underrepresented in the world of computer-assisted reasoning. This workshop aims to bring together researchers working on these subjects, to assess the current state of the art and identify open problems and future research directions. The main topics may be listed (non-exhaustively) as follows: algorithms that may be used in computer-assisted treatment of diagrams, treatment of diagrams in existing computer algebra systems, formal developments related to diagrams and category theory in proof-assistants, user interfaces and graphics for categorical diagrams. There will be space for talks presenting original work, work in progress, applications, survey of previous works. We plan to provide sufficient time for discussions. Details on paper submission will be given in a further announcement. Papers presented at the workshop will be published on the web site of the workshop and may be selected for submission, in complete and revised form, to a special issue of an international journal, in case their number and quality justify it. Important dates: Submission deadline: send short abstract (title) by 30 June 2009 by e-mail to soloviev@irit.fr send either a full paper or an extended abstract by 30 September 2009 Workshop: Friday 16 and Saturday 17 October 2009 Organising/program committee: P. Damphousse (Universite de Tours) Y. Lafont (IML, Universite Aix-Marseille 2) R. Matthes (IRIT, Universite Paul Sabatier, Toulouse) S. Soloviev (IRIT, Universite Paul Sabatier, Toulouse) Local organisation: S. Soloviev R. Matthes A. El Khoury Contact: Sergei Soloviev IRIT University Toulouse-3 118, route de Narbonne, 31062 Toulouse France E-mail: soloviev@irit.fr Tel: (+33) 5 61 55 62 55 Fax: (+33) 5 61 55 62 58 From rrosebru@mta.ca Tue Apr 7 20:03:32 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 07 Apr 2009 20:03:32 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LrKK8-0003Fs-RZ for categories-list@mta.ca; Tue, 07 Apr 2009 20:03:20 -0300 MIME-Version: 1.0 Date: Tue, 7 Apr 2009 12:50:21 -0400 Subject: categories: Re: Where does the term monad come from? From: Zinovy Diskin To: Steve Lack , categories@mta.ca Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: Zinovy Diskin Message-Id: Status: O X-Status: X-Keywords: X-UID: 31 On Fri, Apr 3, 2009 at 12:28 AM, Steve Lack wrote: > > Finitary monads can also be considered on other base categories than Set, > especially on locally finitely presentable ones. > > It is true that vector spaces are the algebras for a finitary monad on Set. > There is no need to restrict to finite-dimensional vector spaces; in fact it > is not true that there is a monad on Set whose algebras are the > finite-dimensional vector spaces. > there is something similar in algebraic logic. The class of locally finite cylindric/polyadic algebras is not a variety and the forgetful functor to Set is not monadic (l.f. means that all relations are of finite arities). In categorical logic (hyperdoctrines), these algebras are considered in many-sorted signatures, in fact, as algebras over graphs, and their theory becomes equational (= the corresponding forgetful functor to Graph is monadic). Probably, it's a general phenomenon wrt specifying finitary objects: by indexing them with finite sets (contexts, supports,arities), we get equational theories over graph-like structures. In a wider (and partly speculative) setting, the shift from classical algebraic to categorical logic is a shift from simple signatures and complex theories to complex signatures and simple theories. In a sense, this is what category theory does wherever it applies to classical problems: it greatly simplifies the logic (and the internal structure), but pays for this by a complex vocabulary (the external structure, interface). A typical example is classical vs. categorical set theories. Thus, a categorical model is a device with a structurally complex interface and simple internal logic. An average user prefers, of course, simple-looking interfaces of classical theories (and eventually has to pay for this choice but it happens later on...). So, for marketing categorical models, it's important to provide good manuals for their complicated interfaces -- what Vaughan just did for monads. Zinovy From rrosebru@mta.ca Wed Apr 8 19:47:06 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 08 Apr 2009 19:47:06 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LrgVQ-0004Ft-9l for categories-list@mta.ca; Wed, 08 Apr 2009 19:44:28 -0300 Date: Wed, 08 Apr 2009 10:44:18 +0200 From: James McKinna MIME-Version: 1.0 To: categories@mta.ca Subject: categories: iBourbaki in constructive type theory? [Fwd: 1 Postdoc and 1 PhD vacancy in the MathWiki project] Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: James McKinna Message-Id: Status: O X-Status: X-Keywords: X-UID: 32 Dear categories-subscribers, Following the recent discussion of iBourbaki and related ideas on this list, you (or your students) may be interested in the following jobs being advertised in our Foundations group at Nijmegen: 1 Postdoc and 1 PhD vacancy in the MathWiki project Please contact Herman Geuvers directly if interested. James McKinna ==================================================== 1 POSTDOC and 1 PHD POSITION in the MathWiki project at Radboud University Nijmegen (NL) http://www.fnds.cs.ru.nl/fndswiki/Vacancies The Institute for Computing and Information Science of the Radboud University Nijmegen (NL) is looking for 2 researchers to work on the NWO project "MathWiki a Web-based Collaborative Authoring Environment for Formal Proofs". The vacancies are: - a POSTDOC for the period of 3 years vacancy number: 62-16-09 - a PHD POSITION for the period of 4 years vacancy number: 62-17-09 AIMS OF THE PROJECT =================== The aim of the MathWiki project is to open up to a wider community the rich collections of knowledge stored in the repositories of proof assistants. To this end we will build a web-based collaborative authoring environment for formal mathematics, the MathWiki system. This system will provide interactive web access through a standardized interface to a number of proof assistants. The MathWiki system will also be a platform for the development of formal proofs within those proof assistants and it will provide high level access (through Wikipedia-like web pages) to their repositories of formalised mathematics. These repositories will reside on the server. In the project we will study and further develop Wiki technology and semantic web technology, all in the context of proof assistant repositories of formalized mathematics. The project thus brings together the open nature of Wiki authoring with expertise in Proof Assistants and Semantic Web technologies to build a new Wiki for mathematics, supporting content creation, search and retrieval. >From the perspective of the ordinary user of mathematics, MathWiki will be important because it will provide high-level mathematical content on the web in a much more coherent and precise way than is available at present. >From the proof assistant user perspective, MathWiki will be important because it will provide an advanced environment for the collaborative authoring of verified mathematics, mediated simply by a web interface. The MathWiki system will be based on our existing experience with proof assistant technology on the web, the "ProofWeb" systems, see http:://prover.cs.ru.nl ----------------------------------------------------------------- Requirements for the PhD student position: - A master's (or equivalent) degree in Computer Science, Mathematics or a related field, with a strong interest in proof assistants and/or semantic web technology (preferably both) - Commitment and a cooperative attitude. - Very good written and oral English skills. Requirements for the Postdoc position: - A PhD in Computer Science, Mathematics, or a related field with expertise in proof assistants and/or semantic web technology (preferably both). - A strong publication record. - Commitment and a cooperative attitude. - Very good written and oral English skills. ----------------------------------------------------------------- Conditions of employment: The PhD students will be employed for a period of 4 years (40 hrs/week). The Postdocs will be employed for a period of 3 years (40 hrs/week). Supervision for the projects will be done by Prof. Dr. Herman Geuvers and Dr. F. Wiedijk Postdoc and PhD student will be appointed by the Radboud University Nijmegen. Both positions shall start before October 1 2009, but preferably earlier. The salary for the PhD position starts at 2042 Euro per month, increasing to 2612 Euro per month in the fourth year. The maximum salary for the Postdoc is 3755 Euro per month (salary scale 10). ----------------------------------------------------------------- Information: For more information, see http://www.fnds.cs.ru.nl/fndswiki/Vacancies. For inquiries about the project and its positions, please contact the project leader Prof. Dr. Herman Geuvers (H.Geuvers@cs.ru.nl, +31 243652603). Interested candidates can ask the project leader for the complete project application text. ----------------------------------------------------------------- Application: Deadline for application is May 1, 2009. Send an application letter with CV and 3 references, mentioning the vacancy number by e-mail to RU Nijmegen, FNWI, P&O mrs. D. Reinders Postbus 9010 6500 GL Nijmegen Netherlands e-mail: pz@science.ru.nl telephone: +31 243652764 From rrosebru@mta.ca Fri Apr 10 09:46:17 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 10 Apr 2009 09:46:17 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LsG4u-00045r-IO for categories-list@mta.ca; Fri, 10 Apr 2009 09:43:28 -0300 Date: Thu, 09 Apr 2009 18:19:47 -0500 (CDT) From: zackluo@j4.com Subject: categories: Monads over a subcategory To: categories MIME-version: 1.0 Content-type: text/plain; charset=us-ascii Content-transfer-encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: zackluo@j4.com Message-Id: Status: O X-Status: X-Keywords: X-UID: 33 There is yet another way to see logic monoid => monad. Definition. Let N be a full subcategory of a category G. A monad (or clone) over N is a pair (K, T) where K is a category with Ob K = Ob N and T is a functor T: K -> G such that for any A, B, C in N (i) K(A, B) = G(A, TB). (ii) f(Tg) = fg for any f in K(A, B) and g in K(B, C) (the order of composition is from left to right). A monad over a category G (in the usual sense) is a monad over the subcategory N = G of G, with K as the Kleisli category and T the right adjoint of the adjunction. A monoid is simply a monad over a singleton (as a subcategory of the category of sets). Examples: Let G = Set be the category of sets. 1. A clone over a singleton is (equivalent to) a monoid. 2. A clone over a finite set is a unitary Menger algebra. 3. A clone over a countably infinite set is simply called a clone. 4. A clone over the subcategory of finite sets is a clone in the classical sense (or a Lawvere theory), which corresponds to a locally finitary clone in the sense of 3 above. 5. A clone over a one-object category is a Kleisli algebra in the sense of E. Manes. For a monad the left algebras (Eilenberg-Moore algebras) represent the smantics (model) and right algebras represent the syntax (logic) of the monad. The theory of right algebras, which is missing from the classical approach to monads, may be applied to study mathematical logic, lambda calculus and recursion theory effectively. From: Clones and Genoids (http://www.algebraic.net/cag/) Zhaohua Luo From rrosebru@mta.ca Sun Apr 12 10:35:25 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 12 Apr 2009 10:35:25 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LsznH-0005aD-JW for categories-list@mta.ca; Sun, 12 Apr 2009 10:32:19 -0300 Message-Id: From: Thorsten Altenkirch To: Steve Lack , Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit Mime-Version: 1.0 (Apple Message framework v930.3) Subject: categories: Re: Where does the term monad come from? Date: Sat, 11 Apr 2009 16:43:13 +0100 Sender: categories@mta.ca Precedence: bulk Reply-To: Thorsten Altenkirch Status: O X-Status: X-Keywords: X-UID: 34 Hi Steve, thank you for addressing the other part of my question. > There was also a second part to the question: > >> >> Btw, I frequently encounter monads in a categories of functors which >> are not endofunctors. An example are finite dimensional vectorspaces >> which can be constructed via a monoid in the category of functors >> FinSet -> Set, here I is the embedding and (x) can be constructed >> from >> the left kan extension and composition. >> The unit is given by the Kronecker delta and join can be constructed >> from Matrix multiplication. Should one call these beasts monads as >> well? Is there a good reference for this type of construction? > > The category of functors from FinSet to Set is equivalent to the > category > of endofunctors of Set which preserve filtered colimits: such > endofunctors > are usually called finitary. Thus a monoid in [FinSet,Set] with > respect to > this tensor product is the same thing as a monad on Set whose > endofunctor > part is finitary: this is called a finitary monad. > > These finitary monads on Set are equivalent to Lawvere theories and > so in > turn to (finitary, single-sorted) varieties. > > Finitary monads can also be considered on other base categories than > Set, > especially on locally finitely presentable ones. > > It is true that vector spaces are the algebras for a finitary monad > on Set. > There is no need to restrict to finite-dimensional vector spaces; in > fact it > is not true that there is a monad on Set whose algebras are the > finite-dimensional vector spaces. I am not sure I completely understand your comments. I guess it may be helpful to be more precise: F : FinSet -> Set F A = Real -> A together with: > eta_A : A -> F A eta a = \ b . if a=b then 1 else 0 (>>=) : F A -> (A -> F B) -> F B v >>= f = \ b. \Sigma a:A.(v a)*(f a b) My notation is inspired by functional programming and naturally as a Computer Scientist I am interested in the constructive content of theorems. This construction only works if the input is decidable (needed for eta) and if we can define Sigma (this certainly works if A is finite). I can see how to lift F to a functor on Sets by using a Kan extension (left ?). In my terminology it may be something like F' : Set -> Set F' X = Sigma A:FinSet. A -> X x F A I suspect my eta and >>= give then rise to a monad on Set? However, I don't see how to do this if the vector spaces are not finite. Btw, I only used this as an example. My question was rather wether people have studied monoids in categories of functors which are not endofunctors. I believe this notion is useful in functional programming and Type Theory as a natural generalisation of the notion of a monad. Cheers, Thorsten This message has been checked for viruses but the contents of an attachment may still contain software viruses, which could damage your computer system: you are advised to perform your own checks. Email communications with the University of Nottingham may be monitored as permitted by UK legislation. From rrosebru@mta.ca Sun Apr 12 10:35:25 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 12 Apr 2009 10:35:25 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lszo8-0005d0-BZ for categories-list@mta.ca; Sun, 12 Apr 2009 10:33:12 -0300 Date: Sun, 12 Apr 2009 11:30:45 +1000 Subject: categories: Re: Where does the term monad come from? From: Steve Lack To: Thorsten Altenkirch , Mime-version: 1.0 Content-type: text/plain; charset="US-ASCII" Content-transfer-encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: Steve Lack Message-Id: Status: O X-Status: X-Keywords: X-UID: 35 Dear Thorsten, I'm not familiar with the notation that you are using, although I can guess what is meant in some cases > > I am not sure I completely understand your comments. I guess it may be > helpful to be more precise: > > F : FinSet -> Set > F A = Real -> A I assume you mean A->Real. It's true that the monad for vector spaces sends a finite set A to R^A, which can be seen as the set of functions from A to R. For a general set A (not necessarily finite) FA is the set of functions from A to R of finite support. Equivalently, FA is the set of formal finite linear combinations of elements of A. > I suspect my eta and >>= give then rise to a monad on Set? However, I > don't see how to do this if the vector spaces are not finite. Yes, this gives a monad on Set whose algebras are vector spaces, not necessarily finite dimensional. I'm not sure what it is you claim to be doing when you "do this". In any case there is a monad on Set whose algebras are vector spaces; there is not a monad on Set whose algebras are finite dimensional vector spaces. You can see this last statement by noting that the category of algebras for a monad on Set is always cocomplete. > > Btw, I only used this as an example. My question was rather wether > people have studied monoids in categories of functors which are not > endofunctors. I believe this notion is useful in functional > programming and Type Theory as a natural generalisation of the notion > of a monad. > Yes, monoids in categories of functors are useful concepts. Of course to define a monoid you need a monoidal structure on the ambient category. There may be many possibilities, and for some of them the corresponding notion of monoid looks more like a monad than for others. For some monoidal structures one should really think of the monoids as not generalizations of monads, but special cases of monads. Your example of finitary monads is a good example. So are operads. There are more examples in the paper "notions of Lawvere theory" available from my home page or as arXiv:0810.2578. Regards, Steve Lack. From rrosebru@mta.ca Mon Apr 13 21:38:17 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 13 Apr 2009 21:38:17 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LtWcj-0007ah-FJ for categories-list@mta.ca; Mon, 13 Apr 2009 21:35:37 -0300 Date: Mon, 13 Apr 2009 11:35:07 -0400 From: "Fred E.J. Linton" To: Subject: categories: Monoids in functor categories [was: Re: Where does the term monad come from?] Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Sender: categories@mta.ca Precedence: bulk Reply-To: "Fred E.J. Linton" Message-Id: Status: O X-Status: X-Keywords: X-UID: 36 On Sun, 12 Apr 2009 09:38:18 AM EDT, Thorsten Altenkirch = , writing to Steve Lack = and , asked, inter alia: = > > ... [snip] ... > = > ... My question was rather wether > people have studied monoids in categories of functors which are not > endofunctors. ... [snip again] ... > = Yes, they have. Two quick examples of such categories, = and what monoids in them boil down to: 1) simplicial sets (cf. Gabriel-Zisman or D.M. Kan for just how this is a functor category) -- here monoids = are "simplicial monoids." Of particular interest, = of course: simplicial groups. 2) modules over a fixed (commutative, say) ring R (a functor category of the form [R, Ab] from the one-object additive category R to the category Ab of abelian groups, consisting of course only of the additive functors) -- here monoids boil down to R-algebras (what in the older van der Waerden terminology were called hypercomplex systems over R). In each instance, of course, one must be careful to specify correctly just which "product" bifunctor on the functor category is to be used in the definition of "monoid" -- in case (2) it's not the usual = (cartesian) product but, rather, a suitable tensor = product one that one wants to be using. I'll let others provide other illustrative examples. Cheers, -- Fred From rrosebru@mta.ca Wed Apr 15 09:25:59 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 15 Apr 2009 09:25:59 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lu49M-0002mE-Tu for categories-list@mta.ca; Wed, 15 Apr 2009 09:23:32 -0300 Date: Wed, 15 Apr 2009 11:35:40 +0200 From: Jaap van Oosten MIME-Version: 1.0 To: categories@mta.ca Subject: categories: PhD vacancy at Nijmegen Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: Jaap van Oosten Message-Id: Status: O X-Status: X-Keywords: X-UID: 37 I forward the message below from Klaas Landsman. ============================================================== I have a vacancy for a PhD student on the project "Topos theory, noncommutative geometry, and quantum logic", starting at a negotiable date in 2009. A detailed description of the project is available at >. The position is for four years (with an initial one-year appointment subject to extension to four years on satisfactory performance), and is financed by NWO. PhD positions in the Netherlands carry a decent salary (going up from about 2000 euro per month gross initially to about 3000 euro in the final year), and the local group of PhD students in mathematics and mathematical physics is exceptionally nice. Moreover, the mathematical physics group at Nijmegen is embedded in the GQT-cluster (see >), officially called "The Fellowship of Geometry and Quantum Theory." Relevant to this project, among its other members are Ieke Moerdijk and Gunther Cornelissen. Applications are welcomed by May 15, 2009, using snail mail (accompanied by a very brief email to to announce your application) to Prof.dr. N.P. Landsman, Radboud Universiteit Nijmegen, Faculty of Science, IMAPP, Heyendaalseweg 135, 6525 AJ NIJMEGEN, THE NETHERLANDS. Please include a letter of motivation, a copy or draft of of your M.Sc. Thesis, a list of courses (including marks), and the names and email addresses of two or three senior academics who could provide information about yourself. Please do not ask them to send references yourself. Female mathematicians or theoretical physicists (with a strong background in mathematics) are especially encouraged to apply. From rrosebru@mta.ca Wed Apr 15 09:25:59 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 15 Apr 2009 09:25:59 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lu48n-0002jZ-5C for categories-list@mta.ca; Wed, 15 Apr 2009 09:22:57 -0300 Date: Tue, 14 Apr 2009 18:53:37 -0400 (EDT) From: larry moss To: larry moss Subject: categories: Workshop on Quantum Logic Insipred by Quantum Computation MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: categories@mta.ca Precedence: bulk Reply-To: larry moss Message-Id: Status: O X-Status: X-Keywords: X-UID: 38 There will be an informal workshop on Quantum Logic Insipred by Quantum Computation at Indiana University, May 11-12. Our aim is to organize a small workshop that would bring together people who are developing new areas of logic coming from quantum computation, and also people who are interested in related projects coming from areas of philosophical logic, mathematics, and theoretical computer science. Information on speakers may be found at www.indiana.edu/~iulg/qliqc. There is no formal registration and all are welcome, but we would appreciate knowing ahead if you plan to come. From rrosebru@mta.ca Wed Apr 15 14:53:59 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 15 Apr 2009 14:53:59 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lu9HX-0006fq-Of for categories-list@mta.ca; Wed, 15 Apr 2009 14:52:19 -0300 From: Hasse Riemann To: Subject: categories: Smooth and proper functors Date: Wed, 15 Apr 2009 13:45:06 +0000 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Sender: categories@mta.ca Precedence: bulk Reply-To: Hasse Riemann Message-Id: Status: O X-Status: X-Keywords: X-UID: 39 =20 Hi category gurus and categorists =20 I have many questions about category theory but i start with one. =20 1> What are smooth functors and proper functors=2C originating in pursuing sta= cks? Both nontechnically and technicaly. =20 I know they are dual to each other and that they are characterized by cohom= ological properties inspired by the proper or smooth base change theorem in algebraic geometry= =2C but what is the relation? (I don't know the statement of the theorems) =20 Finally=2C what are smooth and proper functors good for? Are smooth and proper functors fibrations and cofibrations or Grothendieck = fibrations and Grothendieck op-fibrations in some model categories or derivators? =20 The only thing i could find about smooth and proper functors on internet is= the last entrance in http://golem.ph.utexas.edu/category/2008/01/geometric_representation_theor_= 18.html =20 Best regards Rafael Borowiecki From rrosebru@mta.ca Thu Apr 16 11:11:10 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Apr 2009 11:11:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LuSIC-0000QJ-3b for categories-list@mta.ca; Thu, 16 Apr 2009 11:10:16 -0300 Mime-Version: 1.0 (Apple Message framework v753.1) Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed Content-Transfer-Encoding: quoted-printable From: =?UTF-8?Q?Jonathan_CHICHE_=E9=BD=90=E6=AD=A3=E8=88=AA?= Subject: categories: Re: Smooth and proper functors Date: Thu, 16 Apr 2009 15:46:53 +0200 To: Hasse Riemann , Sender: categories@mta.ca Precedence: bulk Reply-To: =?UTF-8?Q?Jonathan_CHICHE_=E9=BD=90=E6=AD=A3=E8=88=AA?= Message-Id: Status: O X-Status: X-Keywords: X-UID: 40 Hi, The following paper is very clear, I'm currently learning the basics =20 of the subject with it: http://people.math.jussieu.fr/~maltsin/ps/=20 asphbl.ps. It's written in French. Another member of this mailing-=20 list has asked me to translate it in English, I may be able to send =20 you a rough translation in a few weeks. Best, Jonathan Le 15 avr. 09 =E0 15:45, Hasse Riemann a =E9crit : > Hi category gurus and categorists > > > > I have many questions about category theory but i start with one. > > > > 1> > > What are smooth functors and proper functors, originating in =20 > pursuing stacks? > > Both nontechnically and technicaly. > > > > I know they are dual to each other and that they are characterized =20 > by cohomological properties > > inspired by the proper or smooth base change theorem in algebraic =20 > geometry, but what is the relation? > > (I don't know the statement of the theorems) > > > > Finally, what are smooth and proper functors good for? > > Are smooth and proper functors fibrations and cofibrations or =20 > Grothendieck fibrations and > > Grothendieck op-fibrations in some model categories or derivators? > > > > The only thing i could find about smooth and proper functors on =20 > internet is the last entrance in > http://golem.ph.utexas.edu/category/2008/01/=20 > geometric_representation_theor_18.html > > > > Best regards > > Rafael Borowiecki From rrosebru@mta.ca Thu Apr 16 11:11:11 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Apr 2009 11:11:11 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LuSGO-00009Y-4M for categories-list@mta.ca; Thu, 16 Apr 2009 11:08:24 -0300 MIME-Version: 1.0 Date: Wed, 15 Apr 2009 19:44:05 +0100 Subject: categories: Re: Smooth and proper functors From: Andreas Holmstrom To: Hasse Riemann , categories@mta.ca Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: Andreas Holmstrom Message-Id: Status: O X-Status: X-Keywords: X-UID: 41 Hi Rafael, I don't know much about this, but I listened to an excellent talk of Maltsiniotis a few months ago at IHES and posted the scanned notes in a blog post here: http://homotopical.wordpress.com/2009/01/26/maltsinotis-grothendieck-and-homotopical-algebra/ These notes (on page 11-12) contain at least the definition of proper and smooth functors, and the duality statement, so maybe they can be of some limited use. Hopefully other people on this list can provide some more substantial information. Best regards, Andreas Holmstrom 2009/4/15 Hasse Riemann : > > > > Hi category gurus and categorists > > > > I have many questions about category theory but i start with one. > > > > 1> > > What are smooth functors and proper functors, originating in pursuing stacks? > > Both nontechnically and technicaly. > > > > I know they are dual to each other and that they are characterized by cohomological properties > > inspired by the proper or smooth base change theorem in algebraic geometry, but what is the relation? > > (I don't know the statement of the theorems) > > > > Finally, what are smooth and proper functors good for? > > Are smooth and proper functors fibrations and cofibrations or Grothendieck fibrations and > > Grothendieck op-fibrations in some model categories or derivators? > > > > The only thing i could find about smooth and proper functors on internet is the last entrance in > http://golem.ph.utexas.edu/category/2008/01/geometric_representation_theor_18.html > > > > Best regards > > Rafael Borowiecki > > > From rrosebru@mta.ca Thu Apr 16 11:11:23 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Apr 2009 11:11:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LuSJD-0000YZ-Cj for categories-list@mta.ca; Thu, 16 Apr 2009 11:11:19 -0300 Date: Wed, 15 Apr 2009 19:35:34 +0000 (GMT) From: Pierre Cardascia Subject: categories: A theorem from Herrlich and Strecker To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: categories@mta.ca Precedence: bulk Reply-To: Pierre Cardascia Message-Id: Status: O X-Status: X-Keywords: X-UID: 42 Dear Cat=E9goristes,=0A=0AI'm working on an introduction for categorical lo= gic, and I try to avoid using the notion of limit before introducing the no= tion of functor in my work (because limit means limit of functors. Squarely= , we can introduce the limit before, but we don't understand why the limit = is called limit, and limit of what ??).=0ABut I have to introduce the notio= n of categories finitely complete. SO I think about this theorem :=0A=3D=3D= =3D> If C has a terminal object, and a pullback for each pair of arrows wit= h common codomains, then C is finitively complete.=0AI found that without a= ny proof in Goldblatt. Rob Goldblatt just said : "you can find it into such= book from Herrlich and Strecker"... Does somebody has the proof ? Can I us= e this theorem to define complety closed categories instead of working with= limits ? Or does somebody have any way to define complety closed categorie= s without any reference to functors ?=0A=0AThanks !=0A=0APierre CARDASCIA= =0A=0A=0A From rrosebru@mta.ca Thu Apr 16 20:30:55 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Apr 2009 20:30:55 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lub1E-0001U7-5j for categories-list@mta.ca; Thu, 16 Apr 2009 20:29:20 -0300 Date: Thu, 16 Apr 2009 16:22:15 +0100 (BST) From: Jocelyn Paine To: categories@mta.ca Subject: categories: Graphical Category Theory Demonstrations MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: categories@mta.ca Precedence: bulk Reply-To: Jocelyn Paine Message-Id: Status: O X-Status: X-Keywords: X-UID: 43 "If you could commission a computer demonstration of any categorical idea, what would you ask for? Could such demonstrations have helped you, or your students, learn tricky ideas? And, would you be willing to share the visualisations and metaphors that you have devised to explain these ideas to yourself or others?" We've started a thread on this topic at the n-Category Cafe', http://golem.ph.utexas.edu/category/2009/04/graphical_category_theory_demo.html . Anybody interested in using computers to demonstrate concepts from category theory, do please have a look there. I've just added an explanation of the techniques available for delivering demonstrations over the Web, and I mention one - the 3D modelling environment called Alice - that I think will eventually be brilliant for animating category theory, and many other topics besides. Jocelyn Paine http://www.j-paine.org http://www.spreadsheet-parts.org +44 (0)7768 534 091 From rrosebru@mta.ca Fri Apr 17 11:03:59 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 17 Apr 2009 11:03:59 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LuodY-0003xI-EK for categories-list@mta.ca; Fri, 17 Apr 2009 11:01:48 -0300 Date: Fri, 17 Apr 2009 11:22:32 +1000 From: John Bourke MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Category of categories with pullbacks is cartesian closed Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: John Bourke Message-Id: Status: O X-Status: X-Keywords: X-UID: 44 Dear category theorists, I noticed recently that the category whose objects are categories with pullbacks and whose morphisms are pullback preserving functors is cartesian closed. Given a pair of categories with pullbacks A and B, the internal hom [A,B] has objects: pullback preserving functors from A to B, and morphisms: cartesian natural transformations. I have posted a short paper on the arxiv proving this fact: http://arxiv.org/abs/0904.2486 It seems like a fairly natural fact but is not to my knowledge in the literature. I am wondering whether anyone was previously aware of this result, and if so whether it might be mentioned somewhere in the literature? Thanks, John Bourke, University of Sydney. From rrosebru@mta.ca Fri Apr 17 19:30:44 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 17 Apr 2009 19:30:44 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LuwX3-0003b8-LM for categories-list@mta.ca; Fri, 17 Apr 2009 19:27:37 -0300 Date: Fri, 17 Apr 2009 14:06:25 -0400 (EDT) From: Andrew Salch To: categories@mta.ca Subject: categories: pasting along an adjunction MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: categories@mta.ca Precedence: bulk Reply-To: Andrew Salch Message-Id: Status: O X-Status: X-Keywords: X-UID: 45 Let C,D be categories, let F be a functor from C to D, and let G be right adjoint to F. In a recent paper of Connes and Consani, they consider the following "pasting along an adjunction": let E be a category whose object class is the union of the object class of C and the object class of D; and given objects X,Y of E, let the hom-set hom_E(X,Y) be defined as follows: -if X,Y are both in the object class of C, then hom_E(X,Y) = hom_C(X,Y). -if X,Y are both in the object class of D, then hom_E(X,Y) = hom_D(X,Y). -if X is in the object class of C and Y is in the object class of D, then hom_E(X,Y) = hom_C(X,GY) = hom_D(FX,Y). -if X is in the object class of D and Y is in the object class of C, then hom_E(X,Y) is empty. Composition is defined in a straightforward way. When C,D are closed symmetric monoidal categories, then E has a natural closed symmetric monoidal structure as well. Connes and Consani use this categorical pasting to construct schemes over F_1, "the field with one element," and I have worked out some variations and applications of this categorical pasting which produce other useful objects (e.g. algebraic F_1-stacks and derived F_1-stacks, which have some useful number-theoretic as well as homotopy-theoretic properties). I would like to know if this "pasting along an adjunction" is a special case of some more general construction already known to category theory, and if basic properties of pasting along an adjunction have already been worked out and written down somewhere. Thanks, Andrew S. From rrosebru@mta.ca Sat Apr 18 09:40:04 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 18 Apr 2009 09:40:04 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lv9oY-0005gg-Px for categories-list@mta.ca; Sat, 18 Apr 2009 09:38:34 -0300 Mime-Version: 1.0 (Apple Message framework v753.1) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit From: Ross Street Subject: categories: Re: pasting along an adjunction Date: Sat, 18 Apr 2009 14:54:51 +1000 To: Andrew Salch , categories@mta.ca Sender: categories@mta.ca Precedence: bulk Reply-To: Ross Street Message-Id: Status: O X-Status: X-Keywords: X-UID: 46 Dear Andrew There is a bicategory Mod whose objects are categories and whose morphisms are "modules" (also called bimodules, profunctors and distributors). A module from B to A is a functor m : A^op x B --> Set. Modules m : B --> A and n : C --> B are composed using a tensor-product- over-B-like process: see Lawvere's paper: . Every functor g : B --> A gives a module g_* : B --> A taking (a,b) to the set A(a,gb) (which you would write hom_A(a,gb) ). The bicategory Mod has lax colimits (which we call collages because of their gluing- and pasting-like nature). Each single module m : B --> A can be regarded as a diagram in Mod. The collage C of that diagram is the category whose objects are disjointly those of A and of B, morphisms between objects of A are as in A, morphisms between objects of B are as in B, there are no morphisms b --> a, while C(a,b) = m(a,b). There are fully faithful functors i : A --> C and j : B --> C and such cospans A --> C <-- B are precisely the codiscrete cofibrations in Cat. This was important in my paper in Cahiers: Also see which relates to stacks. Your case is the collage of the module g_*. It doesn't matter whether g has an adjoint or not (that simply allows the module to be expressed in two different ways). Regards, Ross On 18/04/2009, at 4:06 AM, Andrew Salch wrote: > Let C,D be categories, let F be a functor from C to D, and let G be > right > adjoint to F. In a recent paper of Connes and Consani, they > consider the > following "pasting along an adjunction": From rrosebru@mta.ca Sat Apr 18 15:29:09 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 18 Apr 2009 15:29:09 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LvFGD-0006KW-9b for categories-list@mta.ca; Sat, 18 Apr 2009 15:27:29 -0300 MIME-Version: 1.0 Date: Sat, 18 Apr 2009 15:42:54 +0200 Subject: categories: Re: pasting along an adjunction From: Urs Schreiber To: Andrew Salch , categories@mta.ca Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: Urs Schreiber Message-Id: Status: O X-Status: X-Keywords: X-UID: 47 On Fri, Apr 17, 2009 at 8:06 PM, Andrew Salch wrote: > In a recent paper of Connes and Consani, > they consider the following "pasting along > an adjunction": [...] > I would like to know if this "pasting along > an adjunction" is a special case of some > more general construction already known > to category theory, and if basic properties > of pasting along an adjunction have already > been worked out and written down somewhere. In section 2.3.1 of "Higher Topos Theory" http://arxiv.org/abs/math.CT/0608040 Jacob Lurie motivates the notion of "inner fibrations" and of Cartesian fibrations of (oo,1)-categories as a generalization of this "pasting" construction. "Pasting" along any bifunctor C^op x D --> Set is the same as having an inner fibration over the interval (which is an arbitrary functor for 1-categories), and the particular "pasting" that you mention, over hom_D(F(-),-) coming from a functor F : C \to D, gives a Cartesian fibration over the interval (top of p. 88, leading over to section 2.4). Best, Urs From rrosebru@mta.ca Tue Apr 21 14:29:56 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 21 Apr 2009 14:29:56 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LwJjn-0004IY-Nf for categories-list@mta.ca; Tue, 21 Apr 2009 14:26:27 -0300 Date: Tue, 21 Apr 2009 15:47:32 +0100 From: Steve Vickers MIME-Version: 1.0 To: Categories , constructivenews@googlegroups.com Subject: categories: Post-doc position at Birmingham: toposes and quantum theory Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: Steve Vickers Message-Id: Status: O X-Status: X-Keywords: X-UID: 48 Research project: "Applications of geometric logic to topos approaches to quantum physics". [I have already announced a PhD studentship on this project, but a 3 yr post-doc position has now come available.] I have received EPSRC funding for a 3-year project with a post-doctoral research position and PhD studentship attached, to be conducted in the School of Computer Science in the University of Birmingham, UK, and to start between now and October. I would be very pleased to receive applications. The essential area of expertise for the post-doc is constructive reasoning (with toposes and point-free topology), but the researcher will also need to have or acquire some familiarity with quantum theory. The funding allows for appointment at a high salary grade for a suitably qualified and experienced researcher. The "topos approaches" referred to are those of Isham and Doering (at Imperial) and Heunen, Landsman and Spitters (at Nijmegen). By working internally in suitable toposes, they are able to find (commutative) Gelfand-Naimark spectra for systems that, externally, are non- commutative. The hope is that this might lead to a style of reasoning about quantum systems that, while logically non-classical, is physically classical. My project will look at trying to keep the intuitionistic, topos- valid, internal reasoning within its geometric part. Insofar as this is possible (and there is mounting evidence that substantial amounts of practical mathematics can be done this way), it enables a language of points, stalks, fibres and bundles for the point-free topology involved. It is hoped that this will allow the topos approach to be conducted in terms that are more conceptually transparent (in particular to physicists), but also make it technically palatable to move from the present presheaf toposes to sheaf toposes. Initial work will focus on the geometric content of the Banaschewski/Mulvey account of Gelfand-Naimark duality. Further information can be found on my web site, at http://www.cs.bham.ac.uk/~sjv/geophysics.php#phd I can also supply a more detailed project description on request. Steve Vickers. From rrosebru@mta.ca Wed Apr 22 21:47:57 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 22 Apr 2009 21:47:57 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lwn6S-0005zd-HV for categories-list@mta.ca; Wed, 22 Apr 2009 21:47:48 -0300 Date: Wed, 22 Apr 2009 15:36:31 -0400 Subject: categories: Preprint available: Close categories vs. closed multicategories From: Oleksandr Manzyuk To: categories@mta.ca Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: Oleksandr Manzyuk Message-Id: Status: O X-Status: X-Keywords: X-UID: 49 Dear category theorists, May I draw to your attention my paper "Close categories vs. closed multicategories" available at http://arxiv.org/abs/0904.3137 In the paper I prove that the 2-category of closed categories of Eilenberg and Kelly is equivalent to a suitable full 2-subcategory of the 2-category of closed multicategories. Comments are welcome! Best, Oleksandr -- "Dealing with failure is easy: Work hard to improve. Success is also easy to handle: You've solved the wrong problem. Work hard to improve." - Alan Perlis From rrosebru@mta.ca Wed Apr 22 21:47:57 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 22 Apr 2009 21:47:57 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lwn3p-0005qo-PJ for categories-list@mta.ca; Wed, 22 Apr 2009 21:45:05 -0300 Date: Tue, 21 Apr 2009 17:37:54 -0700 (PDT) From: John MacDonald To: categories@mta.ca Subject: categories: Re: FMCS 2009 Housing MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: categories@mta.ca Precedence: bulk Reply-To: John MacDonald Message-Id: Status: O X-Status: X-Keywords: X-UID: 50 FMCS 2009 17th Workshop on Foundational Methods in Computer Science University of British Columbia, VANCOUVER, Canada MAY 28th - 31st, 2009 THIRD ANNOUNCEMENT * * * FMCS09 now has a website where one can reserve accommodation online. The address is http://www.pims.math.ca/scientific/general-event/foundational-methods-computer-science-2009 Participants in FMCS 2009 are guaranteed accommodations if reservations are made by April 28, 2009 through the link in the preceding web address. After April 28 the housing office will not guarantee accommodation for our group but will, however, continue to book in the same way as long as space is available. Reservations can be cancelled without penalty until 48 hours before the arrival date so it is to your advantage to book if there is even a slight possibility that you may attend. The next announcement will contain a more complete list of participants so if you are not on the current list of participants and you will or may attend, then please send email to johnm@math.ubc.ca with subject heading FMCS09 - WILL ATTEND or FMCS09 - MAY ATTEND. A preliminary schedule of talks as well as preregistration details will be posted on the website by May 4. In the meantime those who have never attended an FMCS meeting may wish to get an idea of the range of topics by looking at the talks given in 2008 at Halifax appearing in http://www.mscs.dal.ca/~selinger/fmcs2008/ Current List of Participants: Robin Cockett, Computer Science University of Calgary Calgary, Alberta Brett Giles, Computer Science University of Calgary Calgary, Alberta Pieter Hofstra, Mathematics University of Ottawa Ottawa, Ontario Aaron Hunter, Computer Science Simon Fraser University Burnaby, British Columbia Mike Johnson, Mathematics and Computer Science Macquarie University Sydney, Australia John MacDonald, Mathematics University of British Columbia Vancouver, British Columbia Ernie Manes, Mathematics University of Massachusetts Amherst, Massachusetts Phil Mulry, Computer Science Colgate University Hamilton, New York Sean Nichols, Computer Science University of Calgary Calgary, Alberta Vaughan Pratt, Computer Science Stanford University Palo Alto, California Dorette Pronk, Mathematics Dalhousie University Halifax, Nova Scotia Brian Redmond, Computer Science University of Calgary Calgary, Alberta Bob Rosebrugh, Mathematics and Computer Science Mount Allison University Sackville, New Brunswick R A G Seely, Mathematics McGill University Montreal, Quebec Shusaku Tsumoto, Computer Science and Medical Informatics Shimane University Izumo-city, Japan Art Stone, Mathematics Vancouver, British Columbia Hofstra student, Ottawa, Ontario The following paragraphs repeat the information from the first announcement. The Department of Mathematics at the University of British Columbia in cooperation with the Pacific Institute of Mathematical Sciences is hosting the Foundational Methods in Computer Science workshop on May 28th - 31st, 2009, on the University of British Columbia Campus in Vancouver, Canada The workshop is an annual informal meeting intended to bring together researchers in mathematics and computer science. There is a focus on the application of category theory in computer science. However, all those who are interested in category theory or computer science are welcome to attend. The meeting begins with a reception at 6pm in the Ruth Blair room in Walter Gage Towers on the UBC campus on Thursday May 28, 2009. The scientific program starts on May 29, and consists of a day of tutorials aimed at students and newcomers to category theory, as well as a day anda half of research talks. The meeting ends at mid-day on May 31. Research talks There will be some invited presentations, but the majority of the talks are solicited from the participants. If you wish to give a talk please send a title and abstract to johnm@math.ubc.ca. Time slots are limited, so please register early if you would like to be considered for a talk. Graduate student participation is particularly encouraged at FMCS. Registration details will appear in the next announcement. Previous meetings Previous FMCS meetings were held in Pullman (1992), Portland (1993), Vancouver (1994), Kananaskis (1995), Pullman (1996), Portland (1998), Kananaskis (1999), Vancouver (2000), Spokane (2001), Hamilton (2002), Ottawa (2003), Kananaskis (2004), Vancouver (2005), Kananaskis (2006), Hamilton (2007), and Halifax (2008). Organizing committee: Robin Cockett (Calgary) John MacDonald (UBC) Phil Mulry (Colgate) Peter Selinger (Dalhousie) Local Organizer: John MacDonald (UBC) From rrosebru@mta.ca Wed Apr 22 21:47:57 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 22 Apr 2009 21:47:57 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lwn4o-0005uA-Gf for categories-list@mta.ca; Wed, 22 Apr 2009 21:46:06 -0300 MIME-Version: 1.0 To: , "John Bourke" Content-Type: text/plain; charset="utf-8" Date: Wed, 22 Apr 2009 10:29:43 -0400 Subject: categories: re: Category of categories with pullbacks is cartesian closed From: F William Lawvere Content-Transfer-Encoding: quoted-printable Sender: categories@mta.ca Precedence: bulk Reply-To: F William Lawvere Message-Id: Status: RO X-Status: X-Keywords: X-UID: 51 Hi John Interesting result. Especially the way in which the correct 2-structure follows from cartesian closure. An analogous result may be the (exension from=20 posetal domains to categories of) closure under=20 filtered colimits of various shapes. Possibly there is a common generalization to connected limits=20 or colimits ? Bill =20 On Thu 04/16/09 9:22 PM , John Bourke johnb@maths.usyd.edu.au sent: > Dear category theorists, > I noticed recently that the category whose objects are categories with > pullbacks and whose morphisms are pullback preserving functors is cartesi= an > closed. Given a pair of categories with pullbacks A and B, the internal > hom [A,B] has objects: pullback preserving functors from A to B, and > morphisms: cartesian natural transformations.I have posted a short paper = on the arxiv proving this fact: > http://arxiv.org/abs/0904.2486It seems like a fairly natural fact but is = not to my knowledge in the > literature. I am wondering whether anyone was previously aware of this > result, and if so whether it might be mentioned somewhere in the > literature?Thanks, > John Bourke, > University of Sydney. >=20 >=20 >=20 >=20 >=20 >=20 From rrosebru@mta.ca Thu Apr 23 11:43:25 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 23 Apr 2009 11:43:25 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lx07c-00011P-IH for categories-list@mta.ca; Thu, 23 Apr 2009 11:41:52 -0300 Mime-Version: 1.0 (Apple Message framework v753.1) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit From: Philip Scott Subject: categories: Ottawa Workshop on Smooth Structures: Final Call Date: Wed, 22 Apr 2009 23:22:56 -0400 To: Categories list Sender: categories@mta.ca Precedence: bulk Reply-To: Philip Scott Message-Id: Status: RO X-Status: X-Keywords: X-UID: 52 Dear Colleagues: The final schedule and talks for the upcoming Fields Institute Workshop Smooth Structures in Logic, Category Theory, and Physics, May 1--3 at UOttawa are now on the webpage: http://aix1.uottawa.ca/~scpsg/Fields09/Fields09.smoothworkshop.html . If you are planning to come and have not registered yet, please do so and let us know, as we are planning a reception for Saturday night. Registration is on the Fields webpage (or registration can be done here in Ottawa: just let us know). http://www.fields.utoronto.ca/programs/scientific/08-09/ smoothstructures/ We hope to see you here! The Organizers (R. Blute, P. Hofstra, P. Scott, M. Warren) From rrosebru@mta.ca Thu Apr 23 11:43:28 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 23 Apr 2009 11:43:28 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lx097-0001Al-Am for categories-list@mta.ca; Thu, 23 Apr 2009 11:43:25 -0300 MIME-Version: 1.0 Date: Thu, 23 Apr 2009 00:53:13 -0400 Subject: categories: Re: Preprint available: Close categories vs. closed multicategories From: Oleksandr Manzyuk To: pratt@cs.stanford.edu, categories@mta.ca Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Sender: categories@mta.ca Precedence: bulk Reply-To: Oleksandr Manzyuk Message-Id: Status: O X-Status: X-Keywords: X-UID: 53 Dear Vaughan, > Very interesting definition. =C2=A0Do you have an example of a closed cat= egory > that cannot be expanded to a closed monoidal category? =C2=A0If there's o= ne in > your paper then my apologies for overlooking it. Every closed category can be embedded fully faithfully into a closed monoidal category such that the closed structure is preserved; this is due to Laplaza (exact reference in my paper). However, non-monoidal closed categories do occur. I was motivated by the example of A-infinity categories, and frankly, I am not aware of any other non-trivial and non-artificial examples, but I am sure there must be some, it is just my ignorance. Best, Oleksandr --=20 "Dealing with failure is easy: Work hard to improve. Success is also easy to handle: You've solved the wrong problem. Work hard to improve." - Alan Perlis From rrosebru@mta.ca Thu Apr 23 11:44:11 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 23 Apr 2009 11:44:11 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lx09n-0001Dv-TB for categories-list@mta.ca; Thu, 23 Apr 2009 11:44:07 -0300 Mime-Version: 1.0 (Apple Message framework v624) Content-Type: text/plain; charset=US-ASCII; format=flowed Content-Transfer-Encoding: 7bit From: Paul Taylor Subject: categories: Re: Category of categories with pullbacks is cartesian closed Date: Thu, 23 Apr 2009 12:02:36 +0100 To: Categories list Sender: categories@mta.ca Precedence: bulk Reply-To: Paul Taylor Message-Id: Status: RO X-Status: X-Keywords: X-UID: 54 John Bourke > noticed recently that the category whose objects are categories with > pullbacks and whose morphisms are pullback preserving functors is > cartesian closed. Given a pair of categories with pullbacks A and B, > the internal hom [A,B] has objects: pullback preserving functors from > A to B, and morphisms: cartesian natural transformations. > I have posted a short paper on the arxiv proving this fact: > http://arxiv.org/abs/0904.2486 > It seems like a fairly natural fact but is not to my knowledge in the > literature. I am wondering whether anyone was previously aware of > this result, and if so whether it might be mentioned somewhere in the > literature? Along with Francois Lamarche and various other people that I don't clearly recall, I did a lot of work on this idea in the early 1990s. I was based on earlier ideas by Pierre Ageron, Gerard Berry, Yves Diers, Jean-Yves Girard, Peter Johnstone, Andre Joyal, Christian Lair, ... Since the motivations came from either domain theory or generalisations of algebraic theories, the functors that were considered also preserved directed joins or filtered colimits, but these are not relevant to the basic cartesian closed structure. One of the best known categories of posets like this is that of "coherence spaces", which were described in Girard's book "Proofs and Types", which Yves Lafont and I translated. Earlier work by Berry had been the beginning of the search for models of the notion of sequentiality in programming languages. That by Diers had been about a generalisation of algebraic theories to cover the case of fields, with unique disjunction. In the categorical setting, I considered functors that preserve pullbacks. However, Andre Joyal, Francois Lamarche and others considered functors that preserve squares that differ from a pullback by an epi. Then there's the question of whether to preserve equalisers and/or cofiltered limits; for such generalisations I introduced the term "wide pullback". Girard's version had a representation of the function-space. I put this in categorical form by showing that there is a factorisation system in which the "epis" are maps with left adjoints and the "monos" are similar to discrete fibrations. This system is closely related to the Street--Walters "comprehensive" fibration. I described a model whose objects are called "quantitative domains" and which involves permutation groupoids. See www.PaulTaylor.EU/stable/ for my stuff on this subject. There has been renewed interest in some of these things, for which you should look out for words like "species", "shape", "container". Paul Taylor From rrosebru@mta.ca Thu Apr 23 11:44:52 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 23 Apr 2009 11:44:52 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lx0AT-0001Ii-EY for categories-list@mta.ca; Thu, 23 Apr 2009 11:44:49 -0300 From: Hugo.BACARD@unice.fr Date: Thu, 23 Apr 2009 14:02:59 +0200 To: categories@mta.ca Subject: categories: Classifying Space... MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Sender: categories@mta.ca Precedence: bulk Reply-To: Hugo.BACARD@unice.fr Message-Id: Status: O X-Status: X-Keywords: X-UID: 55 Dear category theorists, Sorry for my following stupid questions , but i would like: -Given a monoidal category M , for first assumed to be strict, What kind = of thing do we obtain when we take it's classifying space ?: we take the ner= ve of M and then realising the simplicial sets obtained=20 Explicitely there are theses questions : 1) Does the nerve of M "preserve" (or "reflect") the monoidal structure = of M ? -Is it a monoid in the category of simplicial sets ?=20 -If yes , can we have conditions on a monoid of sSet to be the nerve = of a monoidal category ? I mean, does some kind of "segal condition" ? 2) And What kind of topological spaces of the realization of the nerve -Is it a topological monoid , with some extra structure ?=20 3) And what hapen if M is not strict, or is symmetric, or braided , etc..= .=20 Thank you and sorry if these are completely stupid questions=20 From rrosebru@mta.ca Thu Apr 23 11:46:08 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 23 Apr 2009 11:46:08 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lx0Bg-0001QE-0D for categories-list@mta.ca; Thu, 23 Apr 2009 11:46:04 -0300 Date: Thu, 23 Apr 2009 15:16:55 +0200 From: Carlos Areces Message-Id: <200904231316.n3NDGtc0021300@pluton.loria.fr> Subject: categories: 2nd CFP: Special Issue on Hybrid Logics of the LJ-IGPL Content-type: text/plain Sender: categories@mta.ca Precedence: bulk Reply-To: Carlos Areces Status: O X-Status: X-Keywords: X-UID: 56 *************************************************************** Please excuse for multiple posts and distribute as widely as possible *************************************************************** CALL FOR PAPERS Logic Journal of the IGPL Special issue on Hybrid Logic DEADLINE FOR SUBMISSIONS: 31st May 2009 *************************************************************** Hybrid logic is a branch of modal logic allowing direct reference to worlds/times/states. It is easy to justify interest in hybrid logic on the grounds of applications, as the additional expressive power is very useful. In addition, hybrid-logical machinery improves the behaviour of the underlying modal formalism. For example, it becomes considerably simpler to formulate modal proof systems, and one can prove completeness and interpolation results of a generality that is not available in orthodox modal logic. But more generally, the topic of this special issue is not only standard hybrid-logical machinery (like nominals, satisfaction operators, binders, etc.) but also extensions of modal logic that increase its expressive power in one way or other. For more general background on hybrid logic, and many of the key papers, see the Hybrid Logics homepage (http://hylo.loria.fr/). The special issue will welcome papers in a wide range of topics, including description logic, feature logic, applied modal logics, temporal logic, and labelled deduction. We welcome both theoretical work and work describing systems and applications on hybrid logics, broadly conceived. All submissions will be peer reviewed with respect to the usual journal criteria. Authors are invited to submit original, previously unpublished, research papers written in English. SUBMISSION DETAILS: Papers should not exceed 25 pages including references. Authors are required to prepare their submissions in latex, using the style available at http://hylo.loria.fr/content/SI/2009 Submissions should be sent as .pdf files to areces (at) loria.fr DEADLINE FOR SUBMISSIONS: 31st May 2009 GUEST EDITORS: Carlos Areces (INRIA Nancy Grand Est, areces at loria.fr) Patrick Blackburn (INRIA Nancy Grand Est, blackbur at loria.fr) From rrosebru@mta.ca Fri Apr 24 14:54:27 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 24 Apr 2009 14:54:27 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LxPZB-0001Mj-NO for categories-list@mta.ca; Fri, 24 Apr 2009 14:52:01 -0300 Date: Fri, 24 Apr 2009 09:09:59 +0200 From: David CHEMOUIL To: categories@mta.ca Subject: categories: 2 PhD positions at Onera Toulouse, France Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Sender: categories@mta.ca Precedence: bulk Reply-To: David CHEMOUIL Message-Id: Status: O X-Status: X-Keywords: X-UID: 57 Dear colleagues, would you please forward this e-mail to the students of yours that may be interested by these 3-year PhD positions, proposed at Onera (the French Aerospace Research Center) in Toulouse. Fluency in French *or* in English is mandatory. This subjects are part of a research agenda aiming at helping define and verify large and complex (non-software) systems by using categorical methods as already used in computer science. Best regards, dc --=20 David CHEMOUIL ONERA/DTIM - 2 avenue =C3=89douard Belin - F-31055 Toulouse Tel: +33 (0) 5 6225 2936 - Fax: +33 (0) 5 6225 2593 http://www.onera.fr/staff/david-chemouil FOUNDATIONS FOR SPACE SYSTEMS MODELING =E2=80=94PhD Position at ONERA, Toulouse, France=E2=80=94 =20 =20 Reference: TIS-DTIM2009-02 Application deadline: 2009-05-31 PhD direction: David Chemouil and Virginie Wiels (ONERA), Jean-Paul Bodeveix (IRIT) Contact: David Chemouil ONERA/DTIM 2 avenue =C3=89douard Belin F-31055 Toulouse, France david.chemouil+phd[AT]onera.fr http://www.onera.fr/staff/david-chemouil Subject: The development of space systems (including on-board and ground systems) requires complex engineering and relies upon various disciplines. Until now, the success of such projects has relied upon the quality and experience of engineers as well as on partly-automated development processes. However, although some technical artifacts (e.g: mathematical model of a battery, satellite simulator) are used in such developments, there is still =E2=80=94and mainly=E2=80=94 a huge= amount of paper documentation expressed in natural language. This complexity and diversity of technical means mitigate the validation of a whole space system. Systems model-based development, inspired by similar approaches in the software field, is gaining more and more importance both in the industrial and academic communities. However, the mainstream solutions currently proposed are still unsatisfactory, notably on two major points: =E2=80=93 on one hand, the requirement capt= ure phase and its further refinement towards a formal specification are not well taken into account; =E2=80=93 on the other hand, there is stil= l a lack of formal semantics for architectural models. Both these points have obviously a strong impact on the verification and validation of such systems. The objective of this PhD is to propose relevant concepts for a modeling language addressing these questions. The solution to the first problem may rely on goal-oriented requirement engineering, in the style of KAOS or Tropos methods... The second question will be addressed by devising formal architectural concepts with further (semi-)automated verification in mind. Categorical and/or algebraic approaches, process calculi, model-oriented techniques (such as the B method) will be favoured. Results from software engineering, such as component-based development will be taken into account. The results of the PhD will be backed up by a case study and may be implemented in a software prototype. Candidate profile: MSc in computer science, having followed some =E2=80=9Ctheory=E2=80=9D courses (e.g logic, =CE=BB-calculus, process al= gebra, algebraic specification, B or Z method, etc.). Fluency in English. Gross salary: ranging 1680-2140 EUR /month depending on the profile. 3-year position. Localisation: Onera is the French Aerospace Research Center, affiliated to the French Ministry of Defence. The PhD will be carried out at Toulouse premises. Located in southwest France, Toulouse is a lively city, the 4th most important town in France, and the 2nd one as far as the student population is concerned (see ). A FORMAL SETTING FOR VIEWPOINT COMPOSITION =E2=80=94PhD Position at ONERA, Toulouse, France=E2=80=94 Reference: TIS-DTIM2009-01 Application deadline: 2009-05-31 PhD direction: David Chemouil and Virginie Wiels (ONERA), Sergei Soloviev(IRIT) Contact: David Chemouil ONERA/DTIM 2 avenue =C3=89douard Belin F-31055 Toulouse, France david.chemouil+phd[AT]onera.fr http://www.onera.fr/staff/david-chemouil Subject: A current approach to the engineering of systems (software systems, sociotechnical systems, systems of systems, etc.) consists in modeling them, during specification and design phases, according to various viewpoints. This approach aims at helping engineers handle extremely complex systems by providing them with a view corresponding to their sole business (e.g air or space regulation, dependability, realtime performance, etc.), abstracting from other concerns. Of course, the viewpoints that are useful may strongly vary on the context, henceforth entailing the recurring definition of new multi-view modeling languages. This definition is usually ad-hoc, which forces language designers either to ignore semantic matters and remain on a purely syntactic definition, or to propose a new formal semantics and demonstrate a certain number of otherwise standard results. The aim of this PhD is to propose a theoretical setting to assist in the creation of multi-view languages out of =E2=80=9Csma= ll=E2=80=9D predefined modeling languages (each one being concerned with only one topic). The major question is then to determine under which conditions it is possible to deduce the formal semantics of the multi-view language from the semantics of its constituent languages. Another way to state this problem is to say that we aim at provding a formal semantics to aspect-oriented metamodeling. The foreseen theoretical solutions essentially rely on categorical approaches: algebraic speci- fication, theory of institutions, graph grammars, etc. The results of the PhD will be backed up by a case study and may be implemented in a software prototype. Candidate profile: MSc in computer science or in logic, having followed some =E2=80=9Ctheory=E2=80=9D courses (e.g logic, =CE=BB-ca= lculus, process algebra, algebraic specification, etc.). Fluency in English. Gross salary: ranging 1680-2140 EUR /month depending on the profile. 3-year position. Localisation: Onera is the French Aerospace Research Center, affiliated to the French Ministry of Defence. The PhD will be carried out at Toulouse premises. Located in southwest France, Toulouse is a lively city, the 4th most important town in France, and the 2nd one as far as the student population is concerned (see ). From rrosebru@mta.ca Fri Apr 24 14:54:27 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 24 Apr 2009 14:54:27 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LxPZg-0001Ox-Bo for categories-list@mta.ca; Fri, 24 Apr 2009 14:52:32 -0300 Date: Fri, 24 Apr 2009 12:13:32 -0400 (EDT) From: Robert Seely To: Categories List Subject: categories: Workshop at McGill, 18 June 09 MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: categories@mta.ca Precedence: bulk Reply-To: Robert Seely Message-Id: Status: O X-Status: X-Keywords: X-UID: 58 Workshop announcement: 18 June 2009 at Burnside Hall, McGill, Montreal. We wish to announce a one-day workshop seminar, a "prequel" to the Makkaifest which is being held at CRM 19-20 June (with a reception on the evening of 18 June). http://www.crm.umontreal.ca/Makkaifest09/ http://www.crm.umontreal.ca/Makkaifest09/index_e.php This workshop is intended for the participants of the Makkaifest, and will end in time to allow attendance at the reception at 6pm. If you are interested in attending the workshop, please let us (Phil Scott or Robert Seely) know so we can book a room of suitable size. Any participants who would like to give a talk in the workshop should contact one of the workshop organisers (Phil Scott or Robert Seely) *as soon as possible*, as places are very limited. The themes of the workshop are those of the Makkaifest itself, and should reflect the interests shown by Michael Makkai over his career. We will announce a schedule of talks later, probably late May or early June. We expect the talks to start around 9am and to finish by 5pm, so there'll be time to get to the reception at CRM at 6pm. The workshop will be an informal affair - there will be no registration, and no "refreshments" (people can buy their own coffee, and lunch will be "dutch treat"). Phil Scott (phil@site.uottawa.ca) Robert Seely (rags@math.mcgill.ca) -- From rrosebru@mta.ca Fri Apr 24 19:54:17 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 24 Apr 2009 19:54:17 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LxUGJ-0004g2-Ph for categories-list@mta.ca; Fri, 24 Apr 2009 19:52:51 -0300 Date: Fri, 24 Apr 2009 20:25:21 +0100 (BST) From: Tom Leinster To: Hugo.BACARD@unice.fr, categories@mta.ca Subject: categories: Re: Classifying Space... MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: categories@mta.ca Precedence: bulk Reply-To: Tom Leinster Message-Id: Status: O X-Status: X-Keywords: X-UID: 59 Dear Hugo, Your question involves the functors N | | Cat -----> SSet ------> Top (nerve and geometric realization) and their composite, the classifying space functor B. 1. The nerve functor N has a left adjoint, so in particular it preserves finite products. Hence if M is a monoid in Cat (i.e. a strict monoidal category) then N(M) is, in a natural way, a monoid in SSet. 2. It's also true, though not totally obvious, that the geometric realization functor | | preserves finite products. So if X is a monoid in SSet then |X| is a topological monoid. 3. Putting these together, if M is a strict monoidal category then its classifying space B(M) is a topological monoid. If M is a non-strict monoidal category then B(M) is not necessarily a topological monoid in a natural way, but it is a "homotopy topological monoid" in any of several accepted senses. For instance, it is a Delta-space in the sense of Segal, and an A_infinity-space in the sense of Stasheff (although that doesn't deal satisfactorily with the unit). Similarly, if M is a symmetric monoidal category then B(M) is a "homotopy topological commutative monoid", e.g. a Gamma-space or an E_infinity space. Best wishes, Tom On Thu, 23 Apr 2009, Hugo.BACARD@unice.fr wrote: > > > Dear category theorists, > > > Sorry for my following stupid questions , but i would like: > > -Given a monoidal category M , for first assumed to be strict, What kind of > thing do we obtain when we take it's classifying space ?: we take the nerve of M > and then realising the simplicial sets obtained > > Explicitely there are theses questions : > > 1) Does the nerve of M "preserve" (or "reflect") the monoidal structure of M ? > > -Is it a monoid in the category of simplicial sets ? > -If yes , can we have conditions on a monoid of sSet to be the nerve of a > monoidal category ? I mean, does some kind of "segal condition" ? > > > > 2) And What kind of topological spaces of the realization of the nerve > -Is it a topological monoid , with some extra structure ? > > > 3) And what hapen if M is not strict, or is symmetric, or braided , etc... > > > > > > Thank you and sorry if these are completely stupid questions > > > > > > > > > From rrosebru@mta.ca Sat Apr 25 15:52:10 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 25 Apr 2009 15:52:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lxmw4-0002vV-Bv for categories-list@mta.ca; Sat, 25 Apr 2009 15:49:12 -0300 From: Robert L Knighten MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Date: Fri, 24 Apr 2009 23:54:54 -0700 To: Tom Leinster , Hugo.BACARD@unice.fr, categories@mta.ca Subject: categories: Re: Classifying Space... Sender: categories@mta.ca Precedence: bulk Reply-To: Robert L Knighten Message-Id: Status: O X-Status: X-Keywords: X-UID: 60 Tom Leinster writes: > Dear Hugo, > > Your question involves the functors > > N | | > Cat -----> SSet ------> Top > > (nerve and geometric realization) and their composite, the classifying > space functor B. > > 1. The nerve functor N has a left adjoint, so in particular it preserves > finite products. Hence if M is a monoid in Cat (i.e. a strict monoidal > category) then N(M) is, in a natural way, a monoid in SSet. > > 2. It's also true, though not totally obvious, that the geometric > realization functor | | preserves finite products. So if X is a monoid > in SSet then |X| is a topological monoid. A small issue - this works provided the destination of the geometric realization is the category of compactly generated Hausdorff spaces. Otherwise, as in Milnor's original paper, there are limitation on the simplicial sets involved. -- Bob -- Robert L. Knighten RLK@knighten.org From rrosebru@mta.ca Mon Apr 27 20:06:08 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 27 Apr 2009 20:08:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lxmw4-0002vV-Bv for categories-list@mta.ca; Sat, 25 Apr 2009 15:49:12 -0300 MIME-Version: 1.0 To: "Tom Leinster" , , , "Robert L Knighten" Content-Type: text/plain; charset="utf-8" Date: Mon, 27 Apr 2009 17:27:29 -0400 Subject: categories: Re: Classifying Space... From: F William Lawvere Content-Transfer-Encoding: quoted-printable Sender: categories@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 62 The small issue cited by Bob has further ramifications. Actually the geometric realization functor "should" preserve all finite limits. That is probably a main reason (along with the lack of function spaces) for the gradual demise=20 of Top (and its relatives like Locales) as the default model=20 of cohesion. A functor with a right adjoint and preserving finite limits is the inverse image of a geometric morphism of toposes, provided it is between toposes. Can Top be reasonably=20 replaced by a topos that will serve all purposes of algebraic topology, functional analysis, etc ? Yes, as=20 Peter Johnstone showed some years ago, using in fact=20 a well-known monoid as site. The issue for simplicial realization in particular is whether - the unit interval is totally ordered or not and whether -its endpoints are distinct.=20 That is because simplicial sets is precisely the classifying topos for such structures, as was pointed out by Joyal and explained well both in Johnstone and in Mac Lane & Moerdijk. (The second condition justifies the omission of 0 from the Delta site). Johnstone used the monoid of continuous endomaps of the generic convergent sequence to achieve the total order of the unit interval; the internal meaning of the latter is that inside the square there is no subobject containing both solid triangles. No sheaf, that is. But Peter achieved that solution after detailed study led him to reject=20 another model, proposed by several people whose=20 geometric intuition does not include Peano curves, undecidable statements, etc (e.g., me). That proposal, namely that the basic figures of topology are continuous curves, had to be rejected=20 at the then-current level of knowledge because the usual model for the shape of these figures, the unit interval as constructed in=20 traditional set theory, admits far too many endomaps, along which=20 coverings must be stable by pullback; thus too few coverings , too many sheaves. This leads to the reasonable demand for a submonoid of those continuous reparametrizers, containing polynomials and lattice operations but not containing, for example, the coordinates of a=20 Peano curve in the square. That demand is very similar to the one put forth by Grothendieck in=20 his proposal for TAME TOPOLOGY. I pointed out several years ago that Grothendieck's demand is related to the achievements of mathematicians working on so-called O-minimal models, and some=20 of the of them in fact mention Grothendieck's slogan in their discussions. But I am not aware of any publications addressing the issue exposed by Pete= r. Can we now understand by example the kind of monoid desired ? Best wishes Bill On Sat 04/25/09 2:54 AM , Robert L Knighten RLK@knighten.org sent: > Tom Leinster writes: > > Dear Hugo, > > > > Your question involves the functors > > > > N | | > > Cat -----> SSet ------> Top > > > > (nerve and geometric realization) and their composite, the > classifying> space functor B. > > > > 1. The nerve functor N has a left adjoint, so in particular it > preserves> finite products. Hence if M is a monoid in Cat (i.e. a strict > monoidal> category) then N(M) is, in a natural way, a monoid in SSet. > > > > 2. It's also true, though not totally obvious, that the geometric > > realization functor | | preserves finite products. So if X is a > monoid> in SSet then |X| is a topological monoid. >=20 > A small issue - this works provided the destination of the geometric > realization is the category of compactly generated Hausdorff spaces. > Otherwise, as in Milnor's original paper, there are limitation on the > simplicial sets involved. >=20 > -- Bob >=20 > --=20 > Robert L. Knighten > RLK@knighten > .org >=20 >=20 >=20 >=20 >=20 From rrosebru@mta.ca Thu Apr 30 08:55:49 2009 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 30 Apr 2009 08:55:49 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LzUp9-0002C5-Ux for categories-list@mta.ca; Thu, 30 Apr 2009 08:53:08 -0300 Date: Thu, 30 Apr 2009 08:51:37 -0300 (ADT) From: Bob Rosebrugh To: categories Subject: categories: list interruption / gmane archive MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=X-UNKNOWN Content-Transfer-Encoding: QUOTED-PRINTABLE Sender: categories@mta.ca Precedence: bulk Reply-To: Bob Rosebrugh Message-Id: Status: RO X-Status: X-Keywords: X-UID: 63 [From the moderator:] Your moderator will have only intermittent email contact May 3-15, so there will be delays in posting during that period. As noted below, archives of the categories list are now available in a user friendly format. An archive in mail file format will continue to be maintained and available from the list web page: http://www.mta.ca/~cat-dist/categories.html and the link below will be included there soon. ---------- Forwarded message ---------- Date: Wed, 29 Apr 2009 18:26:16 +0200 From: "Emilio [utf-8] Jes=C3=BAs Gallego Arias" To: Bob Rosebrugh Subject: Re: Categories mailing list request for gmane archival. Dear Prof. Rosebrugh, Bob Rosebrugh writes: > Thanks for your interest in the list. Eventually I'll announce the > availability of this more user-friendly archive. It took a while but categories@mta.ca archives starting from 1997 have been successfully imported into gmane. It will take a while until all the messages are correctly sorted by date. The archives are accessible at http://dir.gmane.org/gmane.science.mathematics.categories Merit for the import should go to gmane's administrator, Lars Magne Ingebrigtsen Thanks a lot for your help. Yours sincerely, Emilio Jes=C3=BAs Gallego Arias >> Gmane is a mail to news gateway, so it basically serves two purposes: >> >> - Allows everyone to read and post to mailing lists with a more >> convenient news reader. >> >> - Archives mail in a web-friendly way. >> >> Additionally, Gmane can import list archives so the full posting history >> of the list is available through it. >> >> The archives at http://www.mta.ca/~cat-dist/archive/ would be an >> excellent source for replaying categories@mta.ca history. However the >> Gmane administrators request that the list moderator approves the >> inclusion, see: >> >> http://gmane.org/import.php >> >> Would you approve it?