From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6313 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: Re: property_vs_structure Date: Mon, 11 Oct 2010 15:03:10 +0200 Message-ID: References: ,<20101007010252.EA0FDCF26@mailscan2.ncs.mcgill.ca> ,<20101008004112.BE2B38572@mailscan2.ncs.mcgill.ca> ,<20101008215343.9920DABC2@mailscan2.ncs.mcgill.ca> Reply-To: "George Janelidze" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1286828041 18006 80.91.229.12 (11 Oct 2010 20:14:01 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 11 Oct 2010 20:14:01 +0000 (UTC) To: "Eduardo J. Dubuc" , , categories@mta.ca> Original-X-From: majordomo@mlist.mta.ca Mon Oct 11 22:13:59 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P5Okw-0006db-HH for gsmc-categories@m.gmane.org; Mon, 11 Oct 2010 22:13:59 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:35790) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P5Ok9-00071j-16; Mon, 11 Oct 2010 17:13:09 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P5Ok5-0000b6-Rf for categories-list@mlist.mta.ca; Mon, 11 Oct 2010 17:13:05 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6313 Archived-At: Dear All, Apologizing to those who heard this many times, I would like to say - sin= ce we are talking about "property_vs_structure" for covering maps: What I called "Galois structures" more than 20 years ago were exactly the structures needed to define covering morphisms in general categories. It = is well known (although it is not clear what it really means!) that Poincare= 's first ideas about covering maps were inspired by Galois theory, and so algebra was "there" even before topology. Having also in mind Grothendiec= k's work, topos-theoretic developments, and Magid's work, I am certainly not original in saying that covering maps should belong to category theory rather than to topology. A Galois structure (say, on a category C) essentially consists of three ingredients (although various modifications are possible): (i) A functor I : C ---> X. It should better have a right adjoint, and it= is wonderful if it is semi-left-exact or, which is almost the same in a sens= e, if it is a fibration. But relative versions of these conditions involving= F below are also good. (ii) A class F of morphisms in C. All covering morphisms we are going to define will be inside F. (iii) Another class E of morphisms in C. Although usually I do not mentio= n it separately because I prefer to define it as the class of effective F-descent morphisms. According to the terminology, Marta and Eduardo used= in their messages, E should now be called a "trivialization structure". For a given Galois structure, the covering morphisms are defined as in my papers, and I would repeat the question about "property vs structure" for covering morphisms as follows: Given a category C, where the concept of a covering seems to be important= , do we want to fix "the best" Galois structure, or we should consider several/many such structures? It is certainly a matter of taste, but to feel the taste one surely needs examples. In my opinion the ones listed below are especially important; t= hey were investigated together with several people you know, whom I was very honoured to work with. Example 1. C =3D the opposite category of commutative rings. Here we have= a very good candidate, which is: X =3D the category of Stone spaces; I : C = ---> X =3D the Boolean spectrum functor; F =3D the class of all morphisms in C= ; E =3D the class of all effective descent morphisms in C. In this case covering morphisms are the same what A. R. Magid called componentially locally strongly separable algebras (considering an R-algebra A as a morphism A -= --> R in C), and are THE most general algebras for which he developed his "separable Galois theory" presented in [A. R. Magid, The separable Galois theory of commutative rings, Marcel Dekker, 1974]. Example 2. C =3D the category of locally connected topological spaces. He= re again, we seem to have "the best candidate". It is: X =3D the category of sets; I : C ---> X =3D the functor sending spaces to the sets of their connected components; F =3D the class of local homeomorphisms of locally connected spaces; F =3D the class of surjective maps from E. The covering morphisms are then the same the covering morphisms of locally connected spaces in the usual sense. Example 3. C is a locally connected topos. This essentially generalizes t= he previous example and everything happens as there (although here F is the class of all morphisms and E the class of all epimorphisms in C of course= ). Marta knows much more than I do about this example and its connections wi= th other topos-theoretic constructions; my only contribution is the short pa= per [G. Janelidze, A note on Barr-Diaconescu covering theory, Contemporary Mathematics 131, 3, 1992, 121-124]. Example 4. C =3D Fam(A) (or FiniteFam(A)), where A is an arbitrary catego= ry with terminal object and "multi-pullbacks" (which simply means that C has pullbacks). This is a further generalization of the same thing, and everything can be repeated, but instead of "epimorphism" we should say "effective descent morphisms" (which is the same thing in the case of a topos). There are many non-topos-theoretic important special cases. For instance if C is the category of all (small) categories, then the coverin= g morphisms are as they should be, that is functors that are discrete fibrations and discrete opfibrations at the same time (this observation i= s due to Steve Lack, although Steve never published it). If C is the catego= ry of all (small) groupoids, then this becomes even nicer since the discrete fibrations of groupoids are the same as discrete opfibrations, are Ronnie Brown often tells us how nicely can they be used in homotopy theory... Example 5. C =3D the category of compact Hausdorff spaces. Here "the best candidate" seems to be: X =3D the category of Stone spaces; I : C ---> X sending compact Hausdorff spaces to the Stone spaces of their connected components; F =3D the class of all morphisms in C; E =3D the class of all morphisms in C that are surjections. As shown in [A. Carboni, G. Janelidz= e, G. M. Kelly, and R. Par=E9, On localization and stabilization of factoriz= ation systems, Applied Categorical Structures 5, 1997, 1-58], the covering morphisms here are the same as light maps in the sense of Eilenberg and Whyburn. Example 6. C =3D the category of simplicial sets. Since it is a category = of the form Fam(A), Example 4 can be used. However, [R. Brown and G. Janelid= ze, Galois theory of second order covering maps of simplicial sets, Journal o= f Pure and Applied Algebra 135, 1999, 23-31] gives a Galois structure that produces a larger (new) class of covering morphisms. In that structure X = is the category of groupoids; I : C ---> X the fundamental groupoid functor,= F the class of Kan fibrations, and E the class of surjective Kan fibrations= . Another "simplicial Galois theory" is presented in [M. Grandis and G. Janelidze, Galois theory of simplicial complexes, Topology and its Applications 132, 3, 2003, 281-289]. Example 7. C =3D the category of groups. The "most classical" candidate w= ould be: X =3D the category of abelian groups; I : C ---> X =3D the abelianiza= tion functor; E =3D F =3D the class of group epimorphisms. In this case the co= vering morphisms are the same as central extensions. This was my first example o= f a "very-non-Grothendieck Galois theory". A bit later I realized that C can = be replaced with any variety of groups with multiple operators in the sense = of [P. J. Higgins, Groups with multiple operators, Proc. London Math. Soc. (3)6, 1956, 366-416] and X with any subvariety in C - and then we get central extensions relative to a subvariety in the sense of A. Fr=F6hlich= 's school (see e. g. [A. Fr=F6hlich, Baer-invariants of algebras, Trans. AMS= 109, 1963, 221-244], [A. S.-T. Lue, Baer-invariants and extensions relative to= a variety, Proc. Cambridge Philos. Soc. 63, 1967, 569-578], [J. Furtado-Coelho, Varieties of W-groups and associated functors, Ph.D. Thes= is, University of London, 1972]). The next step was to get rid of groups completely and Max Kelly and I found out that the crucial property that helps to work with generalized central extensions is congruence modularit= y, and we wrote [G. Janelidze and G. M. Kelly, Galois theory and a general notion of a central extension, Journal of Pure and Applied Algebra 97, 19= 94, 135-161]. Many further results were obtained by Marino Gran, partly in collaboration with Dominique Bourn (see [M. Gran, Applications of categorical Galois theory in universal algebra, Fields Institute Communications 43, 2004, 243-280] and references there for what was done until 2002/3), Tomas Everaert and Tim Van der Linden, and by Tim and Toma= s separately and together. Writing this I feel now bad not to say more abou= t their brilliant results, also involving higher central extensions (see in particular [T. Everaert, M. Gran, and T. Van der Linden, Higher Hopf formulae for homology via Galois theory, Advances in Mathematics 217, 200= 8, 2231-2267]). I shall gladly say more at another occasion. Examples 8-11?. Let C be one of the following three categories: (a) topological spaces; (b) locales; (c) toposes; (d) schemes of algebraic geometry (although (c) is 2-dimensional). I still do not know anything li= ke "the best candidates"... And finally there are trivial examples: (a) C =3D X, with I the identity functor; and (b) X =3D 1. Taking (in both of them) E to be the class of a= ll morphisms in C (and suitable F) we will have: all morphisms in C are covering morphisms in the situation (a), and only isomorphisms are coveri= ng morphisms in the situation (b). In particular, since "the largest" Galois structure is trivial, I would conclude: there is no "the best" Galois structure, and one should rather consider several/many such structures. George [For admin and other information see: http://www.mta.ca/~cat-dist/ ]