May 2010 - Categories - categories.org.au

Research positions
by Maria Manuel Clementino 26 Feb '14

26 Feb '14

5-YEAR RESEARCH POSITIONS The Centre for Mathematics of the University of Coimbra (CMUC) invites applications for five 5-year research positions in all areas of Mathematics. Applicants should have a PhD in Mathematics and at least three years of post-doctoral experience, with an international-level publication record and demonstrated ability to perform independent research. Preference will be given to candidates working in one of the areas in which the members of CMUC are currently active. A very good command of English is required. The appointments will be made for three years, renewable for two more. Successful applicants are expected to develop research within the areas of interest of CMUC. They should also contribute to graduate studies, supervising students from the Department of Mathematics. The basic salary is equivalent to the index 195 of the Portuguese research career (in the order of 42 500 euros a year), in accordance with the Ciência 2008 guidelines http://alfa.fct.mctes.pt/apoios/contratacaodoutorados/edital2008 Benefits include private health insurance and social security. Applicants should send • cover sheet (available from www.mat.uc.pt/cmuc/coversheet.pdf) including names and contact information of three persons who can provide a letter of recommendation • curriculum vitae (publication list included) • a 5-year research proposal (2 pages max), emphasizing their own research goals as pdf files to cmuc(a)mat.uc.pt. The subject of the e-mail message must include the job reference C2008 FCTUC CMUC v15-v19. The deadline for applications is *September 10, 2008.* ------------------------------ More information can be found at www.mat.uc.pt

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separable locale
by Thomas Streicher 17 May '13

17 May '13

Recently rereading Fourman's "Continuous Truth" I came across the term "separable locale" but could nowhere find an explanation. Does it mean a cHa A for which there exists a countable subset B such that ever a in A is the supremum of those b in B with b leq a. This would be the point free account of "second countable", i.e. having a countable basis. Of course, second countable T_) spaces are separable, i.e. have a countable dense set. Is this reading the "usual" one? Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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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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isomorphisms
by Eduardo J. Dubuc 02 Jun '10

02 Jun '10

isomorphims appear not only in examples but are essential also in the theory. For example Grothendieck defines limits and colimits of categories (as universal pseudocones) in SGA4 by means of an isomorphism of categories. Same for toposes. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Isomorphisms of categories
by Peter May 02 Jun '10

02 Jun '10

On 5/30/10 12:50 PM, jim stasheff wrote: > Peter May wrote: >> DeTeXing an exercise I routinely assign, here is >> an example of an isomorphism of categories that is >> not `accidental' in Peter Johnstone's sense and is >> always used in practice as an isomorphism and not >> merely an equivalence. >> >> >> The fundamental theorem of Galois theory: >> >> Let G = Gal(E/F) be the Galois group of a finite >> Galois extension E/F. Define an isomorphism of >> categories between the category of intermediate >> fields F\subset K\subset E and field maps >> K >--> L that fix F pointwise and the category >> of orbits G/H and G-maps between them. >> >> > and an isomprhic category of coverings spaces such that... > > jim No-no, that is in fact the very next exercise: Covering space theory: Requiring covering spaces of a (well-behaved) connected topological space B to be connected, let \sC ov(B) be the category of covering spaces of B and maps over B. If G is the fundamental group of B, then the orbit category of G is {\em equivalent}, not {\em isomorphic}, to \sC ov(B). Sketch the proof. (Hint: use a universal cover of B to construct a skeleton of the category \sC ov(B).) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: terminology
by Ronnie Brown 02 Jun '10

02 Jun '10

Dear André There seems to me to be a tremendous amount of great work going on higher category theory, but when you write ----------------------------------------------------- One lies in the fact that equivalent categories are considered to be the "same", even if [or] when they are not isomorphic. ----------------------------------------- this seems to go against the grain of what I have been doing in groupoids since I decided they were valuable in about 1965! It sounds like the old canard `groupoids reduce to groups', so there must be some confusion in my mind on what you are saying. One thing that took me a while to realise was that it was not enough to study the fundamental groupoid or a fundamental group but one needed to consider intermediate cases, namely the fundamental groupoid on a set of base points chosen according to the geometry at hand. (`Algebraic topology' has not understood this it seems.) The vertices of a groupoid give a spatial component to group theory, a kind of geography, and sometimes, even often, that is needed to model the geometry. So for example it is useful to replace the trefoil group which has 2 generators x,y and one relation x^2=y^3 by the trefoil groupoid which is the double mapping cylinder (homotopy pushout) in groupoids of the two maps Z \to Z, given by squaring and cubing. So we add an extra groupoid generator iota on different vertices which turns x^2 into y^3. This corresponds to the double mapping construction to give a CW-complex. So groupoids give the strict algebra of keeping the information which makes things the same. In higher dimensions we want not just commutative diagrams but control of the ways of filling these diagrams. If the diagram is a pentagon (as we all know does happen) I would want a pentagon as part of the geometry, and the only question is how to deal with multiple compositions of various such objects, and that was the aim of David Jones thesis on Polyhedral T-complexes. The point is that the pieces to be composable have to be all faces but one of a general poyhedral `horn', the process of composing them is the filler of the horn, and the composite of the pieces is the remaining face of the filler. (It was not attempted to do this in category rather than groupoid terms, and that is still a mystery!) So you can see I have long been very sympathetic to using the Kan condition for describing algebraic or structural objects, but find the simplicial approach too awkward (for me, of course; I found the way Nick Ashley coped with that was amazing). I do not want to consider equivalent groupoids the same, as I may want to use the spatial components to describe how they might be glued together. It is partly the old tag of not throwing away information till the last possible moment. On the other hand, some computations are best done at the strict level, rather than the weak one. I mention here the rotations in my paper: ``Higher dimensional group theory'', in {\em Low dimensional topology}, London Math Soc. Lecture Note Series 48 (ed. R. Brown and T.L. Thickstun, Cambridge University Press, 1982), pp. 215-238. (see also a fuller exposition in the new book on Nonabelian algebraic topology), which would seem to be more difficult to write out at the lax level. The fact that the strict calculations imply the existence of certain homotopies is part of the interest. So in the work with Higgins a Kan fibration - from the singular filtered complex of a filtered space to the quotient to give a strict structure - ties in the lax and the strict in a necessary way for the theory and calculations. I am really searching for points of agreement. Best regards Ronnie [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Types Meeting 2010
by Aleksy Schubert 01 Jun '10

01 Jun '10

Types Meeting 2010 Warsaw, 13 - 16 October 2010 First Announcement The 17-th Workshop "Types for Proofs and Programs" will take place in Warsaw, Poland, from October 13 (Wednesday) to October 16 (Saturday). The Types Meeting is a forum to present new and on-going work in all aspects of type theory and its applications, especially in formalized and computer assisted reasoning and computer programming. Invited speakers: * Henk Barendregt; * Yves Bertot; * Pierre-Louis Curien; * Aarne Ranta. Important information concerning accomodation in Warsaw is already available from the conference web page: http://types10.mimuw.edu.pl Details concerning registration, fees, and how to submit a talk will soon be posted there as well. We are looking forward to your participation. The Organizing Committee types10(a)mimuw.edu.pl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Isomorphisms of categories
by Peter May 31 May '10

31 May '10

DeTeXing an exercise I routinely assign, here is an example of an isomorphism of categories that is not `accidental' in Peter Johnstone's sense and is always used in practice as an isomorphism and not merely an equivalence. The fundamental theorem of Galois theory: Let G = Gal(E/F) be the Galois group of a finite Galois extension E/F. Define an isomorphism of categories between the category of intermediate fields F\subset K\subset E and field maps K >--> L that fix F pointwise and the category of orbits G/H and G-maps between them. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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filtered monads? [resent]
by James Borger 31 May '10

31 May '10

Dear category theorists, Does the concept of "filtered monad" exist in the literature? Here are two basic models of what I have in mind. 1. Let C be the category of sets, let F:C->C be the set underlying the free monoid on S, and let F_n(S) be the subset of F(S) consisting of words of length at most n. Then the monad structure map F o F-->F restricts to maps F_m o F_n-->F_{mn}, and F_1 is the identity functor. 2. Let C be the category of R-modules (R a given ring), F(M) is the tensor product R[x] \otimes_R M, and F_n(M) is the sub-R-module M + xM + ... + x^n M of F(M). Then the monad structure map F o F --> F restricts to a map F_m o F_n --> F_{m+n}, and F_0 is the identity functor. So, in the first example, I'd say that the monad F is filtered by the ordered monoid of non-negiative integers under multiplication, and in the second example, it's filtered by that under addition. There seems to be a pretty obvious way of formalizing this, and since many monads in practice come with such a structure, I'd guess that this concept is in the literature, but I didn't find anything on the internet or in the textbooks on my shelf. But perhaps that's because it's not called a "filtered monad" or because it's a special case of a general concept with a completely different name. So, does this concept exist in the literature? I'm writing something about a particular monad with a a filtered structure, and after I define it, I'd like to have the sentence "In the language of [5], the functors F_n endow F with a filtered monad structure." Yours, James Borger ps I'm not at the moment a subscriber to the mailing list, so please cc to me any responses to the list. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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GAMES 2010 - Second call for Papers
by James Worrell 31 May '10

31 May '10

================================================================ GAMES 2010 Annual Workshop of the ESF Networking Programme on Games for Design and Verification September 19 - 23, 2010 St Anne's College, Oxford, UK http://www.comlab.ox.ac.uk/games2010 CALL FOR CONTRIBUTIONS ================================================================ GAMES is the annual workshop of the ESF Network on Games for Design and Verification. This year's GAMES workshop will be held at St. Anne's College Oxford. More information about the games workshop can be found at http://www.comlab.ox.ac.uk/games2010. SCOPE: The scope of the workshop includes the mathematical and algorithmic analysis of finite and infinite games, the interplay of games with automata theory and logic, and applications of games, automata, and logic to the design and verification of computing systems. PROGRAMME: As in previous years, GAMES 2010 will be an informal workshop, without proceedings. Its programme consists of six invited tutorials, contributed talks (30 min) and short presentations (15 min). Contributed talks and short presentations will be selected by the programme committee on the basis of submitted abstracts. These can include work already submitted to other conferences. GAMES 2010 will also feature an open problem session, which will consist of very short (10 min) descriptions of interesting open problems about games. INVITED TUTORIALS: - Roderick Bloem (TU Graz, Austria) - Thomas Colcombet (Liafa, Paris 7, France) - Fedor Fomin (Bergen, Norway) - Paul Goldberg (Liverpool, UK) - Tristan Tomala (HEC, Paris, France) - Wieslaw Zielonka (Liafa, Paris 7, France) SUBMISSIONS: Researchers who would like to present a talk at GAMES 2010 are invited to submit an extended abstract of up to two pages by email to games2010(a)comlab.ox.ac.uk by 1st July 2010. Decisions about acceptance will be made by 1st August 2010. IMPORTANT DATES Paper submission: 1st July 2010 Author notification: 1st August 2010 Workshop: 20-23 September 2010 SUPPORT: We will be able to cover travel and accommodation costs for a limited number of participants (especially students). For information, please contact games2010(a)comlab.ox.ac.uk. There will be no registration fee. LOCATION: The workshop will be held in the Mary Ogilvie Lecture Theatre at St Anne's College, Oxford UK. See http://www.st-annes.ox.ac.uk/ for information about the venue. PROGRAMME COMMITTEE: - Dietmar Berwanger (ENS Cachan) - Mikokaj Bojanczyk (Warsaw) - Patricia Bouyer (ENS Cachan) - Erich Graedel (Aachen) - Stephan Kreutzer (Oxford) - Jean-Francois Raskin (Brussels) - Helmut Veith (TU Vienna) - James Worrell (Oxford) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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