April 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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fundamental localic groupoid?
by David Roberts 02 May '10

02 May '10

Hi all, it is (or should be) well known that the fundamental groupoid of a locally connected, semilocally 1-connected space X can be given a topology such that it is a topological groupoid (composition continuous etc). Without these assumptions on X there are counterexamples to the continuity of composition (c.f. misguided attempts to build a topological fundamental group). I was wondering, though, if there is a localic fundamental groupoid of an arbitrary space. One could presumably pass the the topos Sh(X) and then consider the fundamental groupoid of that, but I was wondering if there was a way to pass directly from the description of the arrows of Pi_1(X) as a set of classes of paths to a locale of classes of paths, and thence to a localic groupoid. My only 'evidence' that this might be the case is that the product in Loc is different to the product in Top, and so this may provide a work around the non-continuity of composition. Thanks, David Roberts [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: fundamental localic groupoid?
by David Roberts 02 May '10

02 May '10

Steven Vickers wrote: > > It's not obvious to me that the path composition can be transferred to that > coequalizer. I think we would like the coequalizer to be stable under > pullback, but that is not always true for locales. Perhaps it is in this > case. > > This is precisely where the construction falls down for a non-semilocally 1-connected space. If we restrict to looking at the automorphisms as a point (which supposedly form a topological group) the multiplication in the group pi_1 x pi_1 -> pi_1 (i.e. concatenating equivalence classes of loops) is not continuous in both variables, even though it is separately continuous (*). This can be traced back to the simple fact that the product of a pair of quotient maps in Top is not necessarily a quotient map. Note that this is the usual product in Top, not the compactly generated product. I presume there is a similar hitch with locales? (*) I learned of the non-continuity of the multiplication map from, among other places, the article J. Brazas, The topological fundamental group and hoop earring spaces, 2009, arXiv:0910.3685 which dashes optimistic earlier constructions. Going back to more general thoughts, given an arbitrary space X, the topos Sh(X) is the classifying topos of some localic groupoid G_X (Joyal-Tierney), which I thought (!) could be interpreted as the fundamental groupoid in nice situations. Here I suppose is my real question: what relation is there between G_X and the (ordinary) fundamental groupoid Pi_1(X)? I suppose a test case could be the Warsaw circle W: Pi_1(W) is equivalent to the discrete groupoid on the set {a,b}, and surely G_W is more than this. One last question: has anyone studied the relation between (strong) shape of a space X and this localic groupoid G_X? David Roberts [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Four problems
by Joyal, André 02 May '10

02 May '10

Jean Benabou has formulated four problems of category theory. They were communicated to a restricted list of peoples, not a private list. I see no serious raisons for not sharing these problems with everyones. Here they are: >Prob1. What conditions must a (small) category C satisfy in order that : >there exists a faithful functor F: C --> G where G is a groupoid? >(Generalized "Mal'cev" conditions) >Prob2. A "little" bit harder, in the same vein. Let C be a (small) category, > S a set of maps of C and P: C --> C[Inv(S)] be the universal functor > which inverts all maps of S. What conditions must the pair (C,S) satisfy >so that the functor >P is faithful? > If P: C --> S is a functor, I denote by V(P) the subcategory of C >which has the same objects and as maps the vertical maps i.e. the f's such >that P(f) is an identity. Let V be a subcategory of a (small) category C. > What conditions >must the pair (C,V) satisfy in order that: >Prob3. There exists a functor P with domain C such that V = V(P) >Prob4. There exists a fibration P with domain C such that V = V(P) Best, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Membranes and monads
by Mike Stay 30 Apr '10

30 Apr '10

There's a design pattern for object-oriented programming called a "membrane" that turns out to be just a pair of adjoint monads; an "identity-preserving membrane" is a monad with an inverse. Do either of these have other names in the category theory literature? -- Mike Stay - metaweta(a)gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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need pointers to literature on category theoretic presentation of co-algebras
by Vasili I. Galchin 29 Apr '10

29 Apr '10

Hello Cat Community, I am currently reading Bart Jaobs et. al presentation on the co-algebra concept: http://citeseer.ist.psu.edu/jacobs97tutorial.html http://www.cs.ru.nl/B.Jacobs/CLG/JacobsCoalgebraIntro.pdf I would pointers to literature that is a more category theoretic presentation of co-algebras. Thank you, Vasili [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Iterating the fundamental groupoid
by Ronnie Brown 29 Apr '10

29 Apr '10

David Roberts writes: ---------------------- In a suitable bicategory of topological groupoids (where internal weak equivalences a la Bunge-Pare or Everaert-Kieboom///-/van der Linden are formally inverted) the topologised fundamental groupoid is equivalent to a groupoid sans topology - in fact it is equivalent to itself where the topology is replaced by the discrete topology. The topologised fundamental groupoid in this way encodes only the 1-type of the space. ------------------------- Isn't this a bit like a sledgehammer to crack a nutshell? The fact is that if X has a universal cover then the topologised fundamental groupoid is such that (s,t): \pi_1(X) \to X \times X is a covering map, so one easily determines that the fundamental groups of \pi_1(X) are determined by those of x. So some new ideas are needed to get at the 2-type of X. I'd like to use this occasion to advertise some of the aims of Grothendieck's writings in the 1980s in the following excerpt, and which may even be relevant to David's question in this thread (??!): In a letter dated 02/05/1983 Alexander Grothendieck wrote: "Don't be surprised by my supposed efficiency in digging out the right kind of notions--I have just been following, rather let myself be pulled ahead, by that very strong thread (roughly: understand non commutative cohomology of topoi!) which I kept trying to sell for about ten or twenty years now, without anyone ready to ``buy'' it, namely to do the work. So finally I got mad and decided to work out at least an outline by myself." This is relevant to `Pursuing Stacks', and to `Derivateurs', for both of which see work by Georges Maltsiniotis. I had found that groupoids were in some sense more powerful than nonabelian cohomology, so in higher dimensions looked for, and eventually found, higher groupoids and colimit results, giving more information than could seemingly be obtained from exact sequences. Ronnie Brown [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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iterating the fundamental groupoid does not work, so.....
by Ronnie Brown 29 Apr '10

29 Apr '10

I would like to make a further point about the topological fundamental groupoid of X. The information on this shows that iterating the fundamental groupoid does NOT lead to higher dimensional information on X. This was one of the facts leading to the Brown-Higgins construction of the homotopy double groupoid \rho(X,X_1,X_0) , where X_0 \subset X_1 \subset X, which in dimension 2 consists of homotopy classes rel vertices of maps of I^2 into X which take the edges of I^2 into X_1 and the vertices into X_0. This does inherit the obvious compositions of squares in two directions to become a strict double groupoid, and with which one can prove a 2-d van Kampen theorem. This was done in 1974, and published in 1978, in the teeth of opposition, which possibly explains why the new nonabelian calculations which resulted have not been generally recognised or taken up. Ronnie Brown [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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