October 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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Reflexive coequalizers
by Richard Garner 06 Nov '06

06 Nov '06

Dear categorists, I have a proof that the indiscrete category functor Set -> Cat preserves reflexive coequalizers which, although straightfoward, uses the explicit description of colimits in Cat. Is this necessary, or can I deduce the result from general principles? [Also, I'm sure this result must appear somewhere but I can't find a reference for it. If anyone knows of one, I'd be grateful.] Many thanks, Richard

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Artin glueing for quasitoposes
by Prof. Peter Johnstone 01 Nov '06

01 Nov '06

Given a subobject of 1 in a topos, it's well known that one can `split' the topos into complementary open and closed subtoposes, and reconstruct the topos (up to equivalence) by applying Artin glueing to these two subtoposes. Hands up, all those of you who thought that the same thing works for quasitoposes ... I thought so! It isn't true, as I have just discovered. Certainly, given a strong subobject U >--> 1 in a quasitopos E, one can construct the `closed complement' of the open subquasitopos E/U, in exactly the same way as one does for a topos: let's denote it by C(U). It's also true that one has a `fringe functor' from E/U to C(U), and that one gets a comparison functor from E to the quasitopos obtained by glueing along this functor (again, the glueing construction works for quasitoposes just as it does for toposes). But, for this comparison to be an equivalence, one needs to know that the inverse image functors E --> E/U and E --> C(U) are (not just jointly faithful, but) jointly isomorphism-reflecting. And that can fail: I have a counterexample in a slice of the quasitopos of Frechet spaces (Elephant, A2.6.4(c). Has anyone noticed this failure before? If so, has anyone actually written it up? Peter Johnstone

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Amendment/Commentary
by F W Lawvere 31 Oct '06

31 Oct '06

Re: Commentary of Adjointness in Foundations in TAC Reprints Dear colleagues, I have taken advantage of the useful possibility for post-publication amendments to add two sentences to my commentary for the TAC Reprint no. 16, Adjointness in Foundations. Matias Menni has made several advances concerning the connection between tractability of proof theory and classical logic of the meta theory. His recent works contain important concepts and precise versions which update the formulations given in my commentary. Thanks to Bob Rosebrugh and Mike Barr. Bill Lawvere ************************************************************ 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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LICS 2007 Call for Workshop Proposals
by Kreutzer + Schweikardt 31 Oct '06

31 Oct '06

LICS 2007 CALL FOR WORKSHOP PROPOSALS The Twenty-Second IEEE Symposium on Logic in Computer Science (LICS 2007) will be held in Wroclaw, Poland, July 10-14, 2007. It will be colocated with two other meetings: the International Colloquium on Automata, Languages, and Programming (ICALP'07) July 9-13, 2007, and also the European Logic Colloquium (ELC 2007), July 14-19. Workshops are planned for July 8, 9 and July 15 (possibly the afternoon of 14th). Detailed information can be found on the LICS 2007 webpage at http://www.informatik.hu-berlin.de/lics/lics07/ LICS workshops have traditionally been an important and exciting part of the program. They introduce either newest research in traditional areas of the LICS community, recent interdisciplinary and applied areas of general theory, or emerging directions that already have some substantial overlap with LICS community interests. Researchers and practitioners are invited to submit proposals for workshops on topics relating logic - broadly construed - to computer science or related fields. Typically, LICS workshops feature a mix of invited speakers and contributed presentations. LICS workshops do not produce formal proceedings. However, in the past there have been special issues of journals based in part on certain LICS workshops. Proposals should include: - A short scientific summary and justification of the proposed topic. This should include a discussion of the particular benefits of the topic to the LICS community. - A discussion of the proposed format and agenda. - Procedures for selecting participants and papers. - Expected number of participants. (This is important!) - Potential invited speakers. - Plans for dissemination (for example, special issues of journals). - For workshops of potential interest to ICALP participants, whether the workshop should be considered as a possible joint LICS/ICALP workshop. - Tell us the timeframe you would prefer: 1 day, 1.5 or 2 days, either before LICS (July 8,9) or after LICS (July 15). Alas, some overlap with either ICALP or ELC is inevitable. Proposals are due Nov. 15, 2006, and should be submitted electronically to: Philip Scott Workshops Chair, LICS 2007 phil(a)site.uottawa.ca Workshops will be chosen by a committee that consists of the LICS General Chair, LICS Workshop Chair, LICS 2007 PC Chair, and LICS 2007 Conference Chair. In the case of potential joint workshops, these will be discussed with colleagues from ICALP. A decision will be made by the end of November.

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University of Oxford: Lectureships in Software Engineering
by Jeremy.Gibbons＠comlab.ox.ac.uk 31 Oct '06

31 Oct '06

UNIVERSITY OF OXFORD Software Engineering Programme Kellogg College THREE UNIVERSITY LECTURERSHIPS IN SOFTWARE ENGINEERING Applications are invited for three new University Lecturerships in Software Engineering. The successful applicants will join the staff of the University's Software Engineering Programme in teaching and researching the application of scientific principles to the development of software systems. The salary will be on a scale up to GBP50,589 per annum. An advanced degree in a related subject, proven teaching ability, and a strong research record - of international standing - are all essential requirements. Applications are particularly welcome from those with expertise in software and systems security, service-oriented architectures, or model-driven development. The appointments will be associated with fellowships at Kellogg College, Oxford and the appointees will be members of the Governing Body of the college. The closing date for applications is 27th November 2006. For further information, including full details of the application procedure and selection criteria, see http://www.softeng.ox.ac.uk/jobs/

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this week's nonsense
by Peter Freyd 30 Oct '06

30 Oct '06

The math advisers from CalTech had a little fun on Friday's episode of "Numb3rs". The Mathematician and the Physicist are looking at a dead person's notebook. They're dead serious. MATHEMATICIAN: it's a category-based approach to I don't know what, but it's a forcasting system. Right?. It's very sophisticated. PHYSICIST: [In awe at what he sees] Oh, no, no. no. This is more than sophisticated. You know what this reminds me of? A topos approach to Bach's music. MATHEMATICIAN: I said: very sophisticated. FBI AGENT: Yes -- you did. Be assured that neither the page we see from the notebook nor the rest of the program had anything to do with categories, topoi or Bach.

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Re: co-smash product (was: categories: Re: Characterization...
by flinton＠wesleyan.edu 29 Oct '06

29 Oct '06

Remarking on George Janelidze's comment, > ... tensor product of commutative rings of A and B without 1 is > nothing but their co-smash product (=the kernel of the canonical > morphism A+B ---> AxB), and therefore Z is the unit object of the smash > product. This observation itself might be infinitely old - simply > because it is simple! Infinitely old the following isn't, but certainly the s.e.s. tensor product --> coproduct --> product made an appearance, for Boolean rngs (boolean algebras w/o unit), in my 1963 Columbia dissertation. Here -- that is, for Boolean rngs -- what's noteworthy is that the tensor product is the same, whether one thinks "as abelian groups, with induced rng structure," "as Z_2-modules, with induced rng structure," or "as the object that represents 'bilinear maps,' i.e., functions on the cartesian product that, for each choice of fixed member of either factor, are homomorphisms on the other factor." [That the third perspective is as valid as the first two comes about because of the idempotence -- xx=x -- of multiplication in Boolean rngs; it surely won't be valid for commutative rngs generally.] On another point, viz., George's comment, > ... co-smash product is associative ... not the case for groups ... what exactly are the relations among group-theoretic co-smash product, group-theoretic tensor product (in the spirit of Hassler Whitney's study from circa 1938 (the very year I was born!)), and group- theoretic commutator constructions? [I recall that Whitney's tensor product of two groups, though defined in terms of representing "bilinear maps," worked out to coincide with the usual tensor product of the groups' abelianizations.] Cheers, -- Fred

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Lawvere-metrics and Banach spaces
by Paul Taylor 29 Oct '06

29 Oct '06

There is a widely cited paper by Bill Lawvere called "Metric spaces, generalised logic and closed catgeories" in which he shows how metric spaces are examples of enriched categories. The enriching structure consists of the nonnegative reals, with "greater than" as the morphisms and addition as the tensor product. Using this one can generalise the notion of metric space by substituting other structures in place of R. An obvious question is - what happens when we follow through this idea for Banach spaces? What becomes of the $\ell_p$ spaces and of dual spaces? Do families of semi-norms naturally fit into this pattern? Paul Taylor

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Characterization of integers as a commutative ring with unit
by Andrej Bauer 27 Oct '06

27 Oct '06

For the purposes of defining the data structure of integers in a Coq-like system, I am looking for an _algebraic_ characterization of integers Z as a commutative ring with unit. (The one-element ring is a ring.) Some possible characterizations which I don't much like: 1) Z is the free group generated by one generator. I want the ring structure, not the group structure. 2) Z is the free ring generated by the semiring of natural numbers. This just translates the problem to characterization of the semiring of natural numbers. 3) Z is the initial non-trivial ring. This is no good because "non-trivial" is an inequality "0 =/= 1" rather than an equality. 4) Let R be the free commutative ring with unit generated by X. Then Z is the image of the homomorphism R --> R which maps X to 0. This is just ugly and there must be something better. I feel like I am missing something obvious. Surely Z appears as a prominent member of the category of commutative rings with unit, does it not? Best regards, Andrej

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