August 2008 - Categories - categories.org.au

Foundations?
by F W Lawvere 03 Sep '10

03 Sep '10

Down with "Foundations"! Up with algebra! Dear Friends, Presumably I am among those who are being "vilified" as "lovers of categories as foundation". By avoiding any precise definition, such a formulation might appeal to the widespread justified boredom induced by the past hundred years of "foundations as justification". Whenever I used the word "foundation" in my writings over the past forty years, I have explicitly rejected that reactionary use of the term and instead used the definition implicit in the work of my teachers Truesdell and Eilenberg. Namely, an important component of mathematical practice is the careful study of historical and contemporary analysis, geometry, etc. to extract the essential recurring concepts and constructions; making those concepts and constructions (such as homomorphism, functional, adjoint functor, etc.) explicit provides powerful guidance for further unified development of all mathematical subjects, old and new. What is the primary tool for such summing up of the essence of ongoing mathematics? Algebra! Nodal points in the progress of this kind of research occur when, as is the case with the finite number of axioms for the metacategory of categories, all that we know so far can be expressed in a single sort of algebra. I am proud to have participated with Eilenberg, Mac Lane, Freyd, and many others, in bringing about the contemporary awareness of Algebra as Category Theory Had it not been for the century of excessive attention given to the alleged possibility that mathematics is inconsistent, with the accompanying degradation of the F-word, we would still be using it in the sense known to the general public: the search for what is "basic". We, who supposedly know the explicit algebra of homomorphisms, functionals, etc. are long remiss in our duty to find ways to utilize those concepts also in guiding high school calculus. Best wishes, Bill Bibliography: - The Category of Categories as a Foundation for Mathematics, La Jolla conference 1965, Springer (1966) - Adjointness in Foundations, Dialectica, (1969), to be reprinted in TAC - Foundations and Applications: Axiomatization and Education, Bulletin of Symbolic Logic, (2003) vol 9, pp 213-224 - Sets for Mathematics, w/ Bob Rosebrugh, Cambridge Univ. Press, (2003) ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************

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KT Chen's smooth CCC
by John Baez 09 Sep '08

09 Sep '08

Hi - Bill Lawvere mentioned that KT Chen had a cartesian closed category of smooth spaces. I've found this very useful in my work on geometry. I kept wanting more properties of this category, so finally my student Alex Hoffnung and I wrote a paper about it: Convenient Categories of Smooth Spaces http://arxiv.org/abs/0807.1704 Abstract: A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace or quotient space of a Chen space is a Chen space, and the space of smooth maps between Chen spaces is again a Chen space. Souriau's "diffeological spaces" share these convenient properties. Here we give a unified treatment of both formalisms. Following ideas of Dubuc, we show that Chen spaces, diffeological spaces, and even simplicial complexes are examples of "concrete sheaves on a concrete site". As a result, the categories of such spaces are locally cartesian closed, with all limits, all colimits, and a weak subobject classifier. For the benefit of differential geometers, our treatment explains most of the category theory we use. In particular, at some point we break down and admit we're dealing with a "quasitopos". Best, jb

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More CT2008 information will be in September
by George Janelidze 01 Sep '08

01 Sep '08

Dear Colleagues, This is just to say, that there will more information on CT2008 in September (since I have not sent in August, as I promised before). I apologize for the delay. George Janelidze

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capturing regularity in languages
by chander pawar 30 Aug '08

30 Aug '08

hi, this is related to theory of formal languages. i would like to know if anyone has some references for , categories of formal languages, in particular "regular languages". how do we capture the notion of 'regularity' in categories? chander

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Re: Resolution; "abutting to"
by Johannes Huebschmann 30 Aug '08

30 Aug '08

Dear All This "situation" reminds me of the following: In 1976 Alexandrov was 80 years old. A. Dold was invited to contribute to the Festschrift, and so he did. Dold wrote his contribution in English and it got translated thereafter into Russian. A little while after the Festschrift had appeared Dold received a letter from a professional translator who had translated his paper from Russian into English. Johannes P. S. I have now found a reference for "spectral sequence abutting to": P. 187 of: J. C. Moore, Cartan's constructions, Colloque analyse et topologie, en l'honneur de Henri Cartan, Ast\'erisque 32--33 (1976), 173--221 On Tue, 26 Aug 2008, Fred E.J. Linton wrote: > Greetings > > Let's just hope none of this creates another situation > like the one Sammy reported facing in a North African > fish restaurant, where his menu offered, among other > local delicacies, "Fried Pimp", the author evidently > having rendered the Arabic word for the fish in question, > actually a mackerel, first into French as "maquereau", > and thence into English as "pimp". > > Rhymes with "shrimp" -- easier to type than "mackerel" -- > so why not? > > Cheers, -- Fred > > ------ Original Message ------ > Received: Mon, 25 Aug 2008 04:13:56 PM EDT > From: Michael Barr <barr(a)math.mcgill.ca> > To: Categories list <categories(a)mta.ca> > Subject: categories: Resolution > >> Thanks to Jonathan Chiche and Johannes Huebschman for the answer to my >> question. First off, according to the online Encyclopedia of Mathematics, >> relatively compact means having compact closure (I had called that >> conditionally compact; neither term is very evocative). >> >> Now to denombrable a l'infini, first Johannes wrote that it meant that the >> one point compactification had a countable basis at the point at infinity. >> Then Jonathan pointed to a '57 paper of M. Zisman that actually defined it >> to mean \sigma-compact. In the context of locally compact spaces, the two >> definitions are easily seen to be equivalent! Since \sigma-compact seems >> to be widely used, I will go with that. >> >> And now let us break off this thread. >> >> Michael >> >> >> > > > > > >

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KT Chen's smooth CCC, a correction
by wlawvere＠buffalo.edu 30 Aug '08

30 Aug '08

In my review of Anders Kock's Synthetic Differential Geometry, Second Edition, there is a wrong statement that I want to correct. (This was in the SIAM REVIEW, vol. 49, No.2 pp 349-350). The statement was that Chen's category does not include the representability of smooth function spaces. But from his paper In Springer Lecture Notes in Mathematics,vol 1174, pp 38-42 it is clear that it does. I thank Anders for pointing out this slip. This is a good opportunity to emphasize that the works of KT Chen and of Alfred Frolicher (that were referred to in the beginning of the above review) contain several contributions of value both to applications and to more topos-theoretic formulations. For example, Frolicher's use of Lemmas by Boman and others reveals how little of the specific parameter "smooth" needs to be given to the very general machinery of adjoint functors and abstact sets in order to obtain smooth infinite dimensional spaces of all kinds. (Namely a suitable topos of actions by only unary operations on the line is fully embedded in the desired topos in such a way that the algebraic theory of n-ary operations that naturally exist in the small one determines the whole algebraic category whose sheaves include the large one.) And Chen's smooth space of piecewise-smooth curves can surely be further applied, as can his special use of convex models for plots. Bill Lawvere

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Smooth categories etc
by R Brown 29 Aug '08

29 Aug '08

Dear Bill and Colleagues, In reply here only to Bill (4), but great interest in the other comments, I entirely agree with the many pointed approach, which was published in my 1967 Proc LMS article. This is relevant also to higher homotopy theory. If you define \pi_n(X,P) where P is now a *set* of base points, then you see this has to have the structure of module over \pi_1(X,P). As an example of its use, consider the map S^n \vee [0,1] \to S^n \vee S^1 which identifies 0 and 1, where the S^n is stuck to [0,1] at 0. Clearly \pi_n of the first space on the set of base points P consisting of 0 and 1 is the free module on one generator over the indiscrete groupoid \I=\pi_1([0,1],P). The main theorem of Higgins and RB implies that \pi_n of the second space at 0 is the free module on one generator over the infinite cyclic group, which itself is obtained from \I by identifying 0 and 1 in the category of groupoids. I know this result can also be obtained using covering space arguments and homology, but I got into groupoids by trying to avoid this detour to covering spaces to describe the fundamental group of S^1. It is a question of finding algebra which models geometry. One of my joint papers, which calculated (using vKT for crossed modules) some \pi_2(X,x) as a module, was rejected by one journal on the grounds that the calculations were too elaborate when `the interest is in the group and not the module'. So much for homological algebra! There is a culture in algebraic topology which neglects the operations of \pi_1; perhaps it seems an encumbrance when there is only one base point, but Henry Whitehead commented in 1957 that the operations fascinated the early workers in homotopy theory. It is amusing to speculate what might be infinite loop space theory, or little cube operads, if you allow many base points! Could it clear up the subjects??!! Answers on a postcard please (joke). A standard lesson in mathematics is that you should forget structure at the latest possible moment. In homotopy theory low dimensional identifications can and usually do affect higher dimensional homotopy invariants. To try and cope with this, we need homotopical functors which carry algebraic information in a range of dimensions, to model how spaces are glued together. This has been the aim of the various Higher Homotopy van Kampen theorems. To handle these algebraic structures one needs category theory ; as one example, I am currently working on a joint paper using fibred and cofibred categories to relate high and low dimensional information on colimits and induced structures. The hope also is that because of the wide interest in deformation, i.e. homotopy, as a means of classification, these tools and methods will have wider implications. Ronnie Dear Ronnie and Colleagues, Your comments are extremely interesting. Thank you very much for raising in so striking a manner the question of the relation between general monoidal structures and cartesian closed structures. Below are some observations which show, I think, that everybody should be interested in this relation because it is manyfold and fruitful. (1) While cartesian closed structures have the virtue of being unique, general monoidal closed structures have the virtue of not being unique. Thus, for example, the cartesian closed presheaf toposes (with their exactness properties and combinatorial truth object) often have a further monoidal closed structure given by Brian Day's convolution with respect to a pro-co-monoidal structure on the site. Cubical as well as simplicial sets have both cartesian and non-cartesian closed structures, and that is 'true', not merely 'convenient'. (2) Another category having both cartesian and non-cartesian monoidal structures is the real interval from zero to infinity with 'x dominates y' as the morphism from x to y. (Actually, this category is derived by collapsing a natural topos of dynamical systems in 'Taking categories seriously' TAC Reprints.) Categories enriched with respect to the non-cartesian structure here (see 'Metric Spaces' TAC reprints) arise every day in analysis and the rich insights of enrichment theory (Functor categories, bi-module composition, free categories, etcetera) should be systematically applied to the advance of analysis and geometry, while on the other hand metric examples inspire further developments of enrichment theory. Cauchy (who never worked on idempotent splitting in ordinary categories and additive categories in the way that Freyd and Karoubi did) does not deserve to have his name brandished as a joke to scare one's uncomprehending colleagues in analysis. The kind of completeness that is inspired by two-sided intervals (unlike the one-sided intervals inaccurately alluded to in common discussions of 'density') indeed reduces to the one attributed to Cauchy in the particular example of Metric Spaces. The author hoped that observation would contribute to the advance of analysis and the development of enrichment theory, not to the supply of buzzwords. In fact, there is an insufficiently known branch of analysis called 'Idempotent Analysis', which deals largely with composition of bi-modules, or more precisely, with the relation between the two closed structures on the infinite interval. Of course, that monoidal category is isomorphic to the unit interval under multiplication (still cartesian closed too) which induces many of the relations between probablility and entropy. (3) Perhaps the most common relation between non-cartesian monoidal categories and cartesian categories arises when a structure such as vector space is interpreted in a cohesive background. I am sticking to my story that cohesive backgrounds are basically cartesian closed, due to the ubiquitous role of diagonal maps and also due to the fact that, for example, bornological vector spaces have an obvious monoidal closed structure, whereas topological vector spaces have none. The rumor that topological vector spaces might have a tensor with an adjoint hom is part of the disinformation that makes functional analysis look more difficult than it is. A more accurate account of the relation between non-Mackey convergence and closed structure can be found in C. Houzel's paper on Grauert finiteness, Mathematische Annalen, vol. 205, 1973, 13-54: essentially, the topological categories are merely enriched in the genuinely monoidal closed bornological ones. Similarly, the idea that not all dual spaces are complete seems to be based on a misguided generality in the notion of Cauchy nets (they should be bounded). (4) Although pointed spaces are somewhat entrenched in algebraic topology, there is an improvement suggested by your own work, Ronnie. Consider the category whose objects are arrows S ---> E where E is a space (object of a cartesian closed cohesive background category) and S is a discrete space. This category is even a topos if the category of E's was, as is the larger category of arrows between general pairs of spaces. The first category is actually an adjoint retract of the second, correcting the discontinuity that arises from the traditional limitation S = 1. Intuitively, in the case where the pair of spaces is a subspace inclusion, the adjoint collapses the subspace to a point if the subspace is connected, but if it is not connected, does not artificially merge its components. There are many applications of this corrected construction of the space which results from 'neglecting' a subspace, both in algebraic topology and in functional analysis, too numerous to discuss here. Bill

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Resolution
by Michael Barr 27 Aug '08

27 Aug '08

Thanks to Jonathan Chiche and Johannes Huebschman for the answer to my question. First off, according to the online Encyclopedia of Mathematics, relatively compact means having compact closure (I had called that conditionally compact; neither term is very evocative). Now to denombrable a l'infini, first Johannes wrote that it meant that the one point compactification had a countable basis at the point at infinity. Then Jonathan pointed to a '57 paper of M. Zisman that actually defined it to mean \sigma-compact. In the context of locally compact spaces, the two definitions are easily seen to be equivalent! Since \sigma-compact seems to be widely used, I will go with that. And now let us break off this thread. Michael

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Asking for more trouble
by Michael Barr 26 Aug '08

26 Aug '08

G is now talking about locally compact spaces and there are two phrases I have not seen. One is "relatively compact". I assume this is the same as what I call "conditionally compact", i.e. having compact closure. The other is "denombrable". The context is "Suppose that the locally compact space $X$ is denombrable a l'infini". Does it mean first countable? I don't want to start a discussion what bad terms these are, I just want to know what they mean. Michael

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Octoberfest 08 - some details
by Robert Seely 26 Aug '08

26 Aug '08

[ Webpage: http://www.math.mcgill.ca/triples/octoberfest08.html ] Centre de Recherche en Theorie des Categories -- Montreal -- Category Theory Research Center -------------------------------------------------------------------------- Category Theory OctoberFest Concordia University, Montreal Saturday - Sunday, October 4 - 5, 2008 -------------------------------------------------------------------------- We invite you to join us in Montreal next October for a weekend meeting in Category Theory, the "not-quite-annual" OctoberFest. As has been the tradition with these meetings, we invite talks from all participants. If you wish to give a talk, send your request along with a short abstract (well before the end of September please) to Robert Seely at the address below. The final schedule of talks will be handed out at the meeting, but a provisional schedule should be available on the webpage, if possible, before the meeting. We will meet in Concordia's Faubourg building ("FG" on the Concordia map) on Saturday morning, October 4th. The FG building is located at 1616 St. Catherine W., corner of St Mathieu (see the Campus map). Lectures will be in room FGB040 in the basement of the FG building. To access this, the easiest way is via the St Mathieu entrance, then go two floors down. We'll provide more explicit instructions on the website soon, and we'll post signs to help on the day. Coffee will be available from 8:30 am. The first talk will be at 9:00. Information re parking: Concordia has an indoor parking garage in the LB (Library) building, 1436 Mackay, near the site of the talks - the daily rate on weekends is only $6. There will be a very modest registration fee to cover refreshments and sundries (details to be specified later). Registration will take place during the morning, before the first talk and during the first coffee break. We plan to arrange a buffet lunch on Saturday at a local Indian restaurant (participants responsable for their own costs). Some recommendations for dinner will be available, but participants are officially on their own(!) in the evening. Please let us know if you intend to join us by sending a short email (before the end of September if possible) to Robert Seely. Below you will find a list of hotels and tourist rooms close to Concordia. We've provided links (on the webpage) to those that ought to be the most suitable. Please note that October is a popular month for visiting Montreal, and early booking is recommended. If you have any further questions, please contact one of us. -------------------------------------------------------------------------- Michael Barr [barr(a)math.mcgill.ca] Robert Raphael [raphael(a)alcor.concordia.ca] Robert Seely [rags(a)math.mcgill.ca] -------------------------------------------------------------------------- Other information to be found on the webpage: Weather: It can be warm, cold, wet, dry ... anything is possible in October. We suggest you check the forecast for the Montreal area Map of the Concordia campus area Parking General map of the area around Concordia -------------------------------------------------------------------------- Hotels: (check the website for links to the closest ones) * Le Chateau Versailles, 1808 Sherbrooke W, 514-933-8111 (1-888-933-8111) * Le Meridien Versailles, 1659 Sherbrooke W, 514-933-8111 (1-800-543-4300) * Hotel du Fort, 1390, rue du Fort, 514-938-8333 (1-800-565-6333) * Hotel Europa Best Western, 1240 rue Drummond, 514-866-6492 (1-800-361-3000) * L'Appartement*, 455 Sherbrooke W, 514-284-3634 * Howard Johnson Plaza*, 475 Sherbrooke W, 514-842-3961 (1-800-842-3961). * Mariott Courtyard*, 410 Sherbrooke W, 514-844-8851 * Holiday Inn*, 420 Sherbrooke W, 514-842-6111 * Hotel du Parc*, 3625 Park Ave, 514-288-6666 Tourist Rooms: (Prices vary depending on choice of single/double room, private/shared bathroom.) * Manoir Ambrose, 3422 Stanley, 514-288-6922, (excellent) * Armor***, 157 Sherbrooke E, 514-285-0140 * Bienvenue B&B*** 3950 Laval, 514-844-5897 (1-800-227-5897) * Casa Bella*, 258 Sherbrooke W, 514-849-2777 * Centre Ville B&B**, 3458 Laval, 514-289-9749 * Pierre***, 169 Sherbrooke E, 514-288-8519 *15 minute walk from Concordia **30 minute walk from Concordia ***45 minute walk from Concordia (Note that using the Metro or bus makes the trip shorter) -- <rags(a)math.mcgill.ca> <www.math.mcgill.ca/rags>

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