August 2006 - 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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Linear--structure or property?
by Michael Barr 04 Sep '06

04 Sep '06

Bill Lawvere uses "linear" for a category enriched over commutative semigroups. Obviously, if the category has finite products, this is a property. What about in the absence of finite products (or sums)? Could you have two (semi)ring structures on the same set with the same associative multiplication? Robin Houston's startling (to me, anyway) proof that a compact *-autonomous category with finite products is linear starts by proving that 0 = 1. Suppose the category has only binary products? Well, I have an example of one that is not linear: Lawvere's category that is the ordered set of real numbers has a compact *-autonomous structure. Tensor is + and internal hom is -. Product is inf and sum is sup, but there are no initial or terminal objects and the category is not linear.

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Martins-Porter paper
by John Baez 31 Aug '06

31 Aug '06

Interesting letter from Grothendieck! Ronnie Brown writes: >A recent arXiv paper by Jo=E3o Faria Martins and Tim Porter >math.QA/0608484 >Title: On Yetter's Invariant and an Extension of the Dijkgraaf-Witten >Invariant to Categorical Groups > > seems also relevant to the theme of the Baez-Shulman notes. Yes; we didn't get into TQFTs at all in our notes, but Martins and Porter use the classifying space of a discrete 2-group (=3D categorical group =3D crossed module =3D gr-category) in their construction, and that's quite relevant. I wrote a little introduction to the Martins-Porter paper and its antecedents at the n-category cafe: http://golem.ph.utexas.edu/category/2006/08/categorifying_the_dijkgraafwit.= html Best, jb

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Classifying spaces
by Peter May 29 Aug '06

29 Aug '06

On colimits of classifying spaces. This topologist may be missing something, but the conclusion seems obviously true, at least in reasonable situations. With a countable system of inclusions of spaces, unless the situation is fairly bizarre, the system will be filtered and we can find a countable cofinal sequence. But finite products commute with sequential colimits in reasonable categories of spaces. Since the usual classifying space of G is constructed as the geometric realization of a simplicial space whose space of q-simplices is G^q, and since geometric realization certainly commutes with sequential colimits, the conclusion seems clear. It is used all the time in algebraic topology, in such familiar examples as BU = colim BU(n), BTop = colim BTop(n), BF = colim BF(n), etc, the last being a system of monoids rather than groups. In the standard classical statement that BU classifies stable complex vector bundles, we are using the first listed special case. While the groups are compact in that case, compactness is not relevant to the argument and fails for the groups Top(n), for example. In more complicated equivariant situations, one has countable systems that are not naturally sequential, but again the conclusion is familiar and in common use. Peter

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classifying functor and colimits
by David Roberts 29 Aug '06

29 Aug '06

Dear category theorists, I have been plagued by the following question: does the classifying space functor commute with (co)limits? In particular, I have a system of compact topological groups G_i indexed by the natural numbers, and a whole lot of inclusions. Is B colim G_i homotopic to colim BG_i ? I have a hint that this should be so in my particular situation (in a letter of Serre to Grothendieck), but I'd like to know how the general case goes. Cheers, ------------------------------------------------------------------------ -- David Roberts School of Mathematical Sciences University of Adelaide SA 5005 ------------------------------------------------------------------------ -- droberts(a)maths.adelaide.edu.au www.maths.adelaide.edu.au/~droberts www.trf.org.au

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Re: lectures on n-categories and cohomology; Grothendieck comments
by Ronnie Brown 28 Aug '06

28 Aug '06

John, in his very nice notes with Shulman, quotes from a letter to Quillen which formed the start of Alexander Grothendieck's `Pursuing Stacks'. Since this explicitly mentions Bangor, it could be useful to readers to read an extract from a later letter to me. I'll put a pdf file on my web site in due course. `Pursuing Stacks' is to be published in Documents Math\'ematiques, with various correspondence as an Appendix, edited by Georges Maltsiniotis. A recent arXiv paper by João Faria Martins and Tim Porter math.QA/0608484 [abs, ps, pdf, other] : Title: On Yetter's Invariant and an Extension of the Dijkgraaf-Witten Invariant to Categorical Groups seems also relevant to the theme of the Baez-Shulman notes. Ronnie Brown Extract from a letter Alexander Grothendieck to Ronnie Brown, 06/ 09/1983 It is all too evident I am not an expert on homotopy theory, and the books I am bold enough to write now on foundational matters are very likely to be looked at as ``rubbish" too by most experts, unless I show up with $\pi_{147}(S^{123}$ as a by-product (whereas it is for the least doubtful I will...). At the very least, you should give me some hints as to the kind of things I could reasonably say in a ``formal note of support", besides how nice it would be to have a better understanding of the foundational matters. This makes me think by the way that (much to my surprise, I confess) I never got a line from Quillen in reply to my long letter from February. I guess since that time he should have gotten that letter, maybe you even gave him a copy time ago if I remember it right. As two letters for me in the Faculty mail got lost lately, it isn't wholly impossible that he did reply and I didn't get it. In case you should know something on this behalf, please tell me. I realize somewhat belatedly that I should apologize for the mistaken impression I got, from a quick glance through the heap of reprints you sent me a year or so ago, and which I somewhat bluntly expressed in my first letter to you I believe - namely that you had little or no background in so-called ``geometry". It would be more accurate, it seems, to say that your background and mine don't overlap too much. My own background has been somewhat moving for the last ten or twelve years, since I withdrew rather abruptly from the mathematical milieu. Thus my interest in the Teichm\"{u}ller (or mapping class) group has developed mainly, in two steps, during the last two years and a half. It came quite as a surprise that you have come to some contact with these groups, too - and I would be quite interested to get a reference on this ``amazing finite presentation" you are speaking of (and I can well imagine it must be tied up with the Mumford-Deligne compactification of the relevant modular multiplicity, whose $\pi_1$ is the group we are looking at). I was under the impression that to give an explicit presentation of the group, rather than of the groupoid, would be kind of inextricable, and it is surely an interesting fact it is not. Still, I am pretty sure for the ``arithmetical'' theory I am interested in, that one just cannot possibly dispense from working with groupoids, rather than just groups. A few times in your letter you stop to ask what of all you're saying would make sense with spaces replaced by topoi, and wondering if it would be a long way to do those things in the wider context. If you are just interested in homotopy types (more accurately, prohomotopy types) of topoi, it seems to me that Artin-Mazur have developed more or less all the machinery needed, in order for any result in semisimplicial homotopy theory, say, to carry over more or less automatically to topoi. This isn't really the most interesting thing they did, but rather what could be considered as the routine part of their work, which they develop by standard semisimplicial homotopy techniques. What they were really after was giving various ``profinite" variants of homotopy types and a formalism of ``profinite completion" of usual (pro )homotopy types, relevant when working with \'etale cohomology of schemes, and using this, stating and proving a few key theorems, a typical one being that for a proper and smooth morphism of schemes $[f]$ and taking profinite completions (of homotopy types) ``prime to the residue characteristics", the theoretical ``homotopy fiber" of the map $[f]$ can be identified with the (prohomotopy type of the) actual schematic geometric fibers of the map $[f]$. It turns out that the algebraic machinery reduces these statements to corresponding statements about cohomology with torsion coefficients (including non-commutative cohomology in dimension 1), which had all been proved in the SGA4 seminar by Artin and me. I think within the next day I am going to read through your preprint ``An introduction to simplicial T-complexes", as you suggested, maybe I'll write again if I have any questions. For the time being, I guess I'll stop. And thank you again very much for your patient help. Very affectionately Alexander ----- Original Message ----- From: "John Baez" <baez(a)math.ucr.edu> To: "categories" <categories(a)mta.ca> Sent: Tuesday, August 22, 2006 7:54 AM Subject: categories: lectures on n-categories and cohomology > Some of you may enjoy this: > > Lectures on n-Categories and Cohomology > John Baez and Michael Shulman > http://arxiv.org/abs/math.CT/0608420 > > The goal of these talks was to explain how cohomology and other > tools of algebraic topology are seen through the lens of n-category > theory. Special topics include nonabelian cohomology, Postnikov > towers, the theory of "n-stuff", and n-categories for n = -1 and -2. > The talks were very informal, and so are these notes. A lengthy > appendix clarifies certain puzzles and ventures into deeper waters > such as higher topos theory. For readers who want more details, > we include an annotated bibliography. > > > > > > > > -- > Internal Virus Database is out-of-date. > Checked by AVG Free Edition. > Version: 7.1.394 / Virus Database: 268.10.4/402 - Release Date: 27/07/2006 > >

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Laws
by Tom Leinster 25 Aug '06

25 Aug '06

Dear category theorists, Here's something that I don't understand. People sometimes talk about algebraic structures "satisfying laws". E.g. let's take groups. Being abelian is a law; it says that the equation xy = yx holds. A group G "satisfies no laws" if whenever X is a set and w, w' are distinct elements of the free group F(X) on X, there exists a homomorphism f: F(X) ---> G such that f(w) and f(w') are distinct. For example, an abelian group cannot satisfy no laws, since you could take X = {x, y}, w = xy, and w' = yx. There are various interesting examples of groups that satisfy no laws. To be rather concrete about it, you could define a "law satisfied by G" to be a triple (X, w, w') consisting of a set X and elements w, w' of F(X), such that every homomorphism F(X) ---> G sends w and w' to the same thing. A law is "trivial" if w = w'. Then "satisfies no laws" means "satisfies only trivial laws". You could then say: given a group G, consider the groups that satisfy all the laws satisfied by G. (E.g. if G is abelian then all such groups will be abelian.) This is going to be a new algebraic theory. What bothers me is that I feel there must be some categorical story I'm missing here. Everything above is very concrete; for instance, it's heavily set-based. What's known about all this? In particular, what's known about the process described in the previous paragraph, whereby any theory T and T-algebra G give rise to a new theory? Thanks, Tom

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Attribution re no membership-respecting morphisms
by Vaughan Pratt 24 Aug '06

24 Aug '06

I've attributed the third saying at http://boole.stanford.edu/dotsigs.html to Mike Barr. If there's an earlier attribution I should be using please let me know. Vaughan Pratt

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The n-Category Cafe
by John Baez 24 Aug '06

24 Aug '06

Dear Categorists - David Corfield, Urs Schreiber and I have started a group blog on math, physics and philosophy called The n-Category Cafe: http://golem.ph.utexas.edu/category/ You're all invited to come over, order an espresso and chat. We'll talk about lots of things besides n-category theory, but that's one of our shared interests, so we'll probably talk a lot about that. One interesting feature of this blog is that it lets you use TeX. Best, jb

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[3WFTop] Third Workshop on Formal Topology. First Announcement.
by Peter Schuster 22 Aug '06

22 Aug '06

3WFTop First Announcement: THIRD WORKSHOP ON FORMAL TOPOLOGY Accademia Galileiana, Padua (Italy) 9-12 May 2007 workshop 7-8 May 2007 tutorials This is the third of a series of successful meetings on the development of Formal Topology and its connections with related approaches. The first two have been held in Padua, 1997, and Venice, 2002. For more information on 3WFTop see http://www.3wftop.math.unipd.it/ What is formal topology When topology is developed in a strictly constructive way, for instance over Martin-Loef's type theory, points cannot be given primitively and the pointfree approach is fundamental. This is the reason why it is called formal. Formal topology has now become an important tool in constructive mathematics. More on formal topology: http://www.3wftop.math.unipd.it/formal-topology.html Invited speakers Invited speakers include Andre' Joyal, Per Martin-Loef and many other prominent scholars: http://www.3wftop.math.unipd.it/invited-speakers.html Tutorials Before the workshop, two days of extensive and coordinated tutorials are planned, given by Bernhard Banaschewski and other pioneers: http://www.3wftop.math.unipd.it/tutorials.html Contacts If you wish to be kept updated with information about 3WFTop, please send an e-mail to: fortop(a)math.lmu.de with KEEP ME UPDATED in the subject. the Scientific Committee Thierry Coquand Giovanni Sambin Peter Schuster

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