Re: property_vs_structure
Dear George, Thank you for reminding us of your old notion of Galois structure and covering morphism in general categories. Although tangentially relevant to the discussion initiated by Eduardo Dubuc, it relates to examples of properties of continuous maps of spaces (or or morphisms of toposes) studied in my book with Jonathon Funk, which may be relevant. I sent you this privately already, but on second thoughts I think it might be useful to make it public. I begin by quoting a paragraph from your posting.
Example 4. C = Fam(A) (or FiniteFam(A)), where A is an arbitrary category 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 covering morphisms are as they should be, that is functors that are discrete fibrations and discrete opfibrations at the same time (this observation is due to Steve Lack, although Steve never published it). If C is the category 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...
The notions of discrete fibration and discrete opfibration are lifted from categories to geometric morphisms of toposes (in M. Bunge and J. Funk, Singular Coverings of Toposes, LNM 1890, Springer, 2006, Chapter 9) relative to the symmetric KZ-monad called M therein for "measures" (M.Bunge and A.Carboni, JPAA 105 (1995) 233-249). They are, respectively, the local homeomorphisms and the complete spreads (singular coverings). A local homeomorphism over a locally connected space E with defining object X is said to be an unramified covering if it is also a complete spread. Unramified coverings generalize covering morphisms over a locally connected space-- if X is a locally constant object of a locally connected space E, then the corresponding local homeomorphism is a complete spread, hence an unramified covering. The class of unramified coverings is strictly larger than the class of locally constant coverings, even over a locally connected space (J. Funk and E.D. Tymchatyn, Unramified maps, J. Geometric Topology 1(3) (2001) 249-280). Under hypotheses of the locally simply connected kind, unramified coverings are locally constant. The larger class of unramified coverings has some nice properties which the class of locally constant coverings fails to have -- for instance, they compose. Moreover, a van Kampen theorem holds not just for the class of locally constant coverings but also for the larger class of unramified coveirngs (M.Bunge and S. Lack, Van Kampen theorems for toposes, Advances in Mathematics 179/2 (2003) 291-317). It is clear from your theory that both classes of morphisms are instances of what you call a Galois structure on the category of (locally connected) topological spaces.
Best wishes, Marta [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Marta, Many thanks, and apologizing for the delay, I am now answering: As you know better than I do, Topos Theory has many aspects and great impact (using these days' expression) on many areas of mathematics. But in this message let me consider only one of its aspects, namely that the (2-)category TOP of toposes can be considered as one of the candidates for 'the right geometric/topological category'. The long list of other possible candidates includes Fam(A) = the category of families of objects of a category A, such that Fam(A) has pullbacks (e.g. every (cocomplete) locally connected topos is such); Loc = the category of locales, Top = the category of topological spaces, CHTop = the category of compact Hausdorff spaces, LaxAlg(T,V) = the categories of lax (T,V)-algebras in the sense of M. M. Clementino, D. Hofmann, and W. Tholen (if T is the ultrafilter monad of Sets, and V = {0,1}, then LaxAlg(T,V) = Top by a theorem of M. Barr), Schemes = the category of schemes in algebraic geometry, CR^o = the opposite category of commutative rings, and many others (I listed only those that will be mentioned below). Each of them certainly has various (subcategories with various) Galois structures with interesting covering morphisms. But essentially only in the cases of CR^o, CHTop, and Fam(A) I have a feeling that the Galois structure I am using (which is just the adjunction with Stone spaces for CR^o and for CHTop, and with sets for Fam(A)) I am using is THE right one. In particular the case of TOP seems to be very interesting, and probably "the answer" would give an answer for Loc, while a "good answer" for Loc might suggest something for TOP. I am not sure I fully understood what you say about unramified coverings versus locally constant coverings. Are you even saying that you found a Galois structure on TOP, or on any subcategory of TOP, whose coverings are exactly the unramified coverings (and the situation is non-trivial in the sense that unramified coverings are not the same as locally constant coverings? That would be wonderful! Independently of that your work on study and comparing what you call local homeomorphisms, complete spreads, and unramified coverings in TOP is absolutely very interesting! And there should be a connection to be understood between it and the work of Maria Manuel Clementino and Dirk Hofmann on similar concepts in LaxAlg(T,V). You say "The notions of discrete fibration and discrete opfibration are lifted from categories to geometric morphisms of toposes (in M. Bunge and J. Funk, Singular Coverings of Toposes, LNM 1890, Springer, 2006, Chapter 9) relative to the symmetric KZ-monad called M therein for "measures" (M.Bunge and A.Carboni, JPAA 105 (1995) 233-249). They are, respectively, the local homeomorphisms and the complete spreads (singular coverings)..." And this is to be compared with the following: Just as for categories, there are discrete fibrations and discrete opfibrations of preorders, and coverings are exactly those that are discrete fibrations and discrete opfibrations at the same time. On the other hand finite preorders are the same as finite topological spaces, and discrete fibrations of finite preorders are the same as the local homeomorphisms of finite topological spaces. This generalizes to the infinite case as follows: Using the ultrafilter convergence, one can define discrete fibrations of T-preorders (=lax T-algebras = T-categories), where T is the ultrafilter monad on the category of sets; and it turns out that: (i) the class of discrete fibrations of T-preorders is not pullback stable; (ii) the pullback stable discrete fibrations of T-preorders are the same as local homeomorphisms of general topological spaces (see [M. M. Clementino, D. Hofmann, and G. Janelidze, Local Homeomorphisms via Ultrafilter Convergence, Proc. AMS 133, 3, 2004, 917-922]). Similarly to the case of T-preorders one can define discrete fibrations and discrete opfibrations of lax (T,V)-algebras (for arbitrary T and V), and this is what I would like to compare with your local homeomorphisms and complete spreads. Of course the categories TOP and LaxAlg(T,V) are so different that the only way to make such a comparison, would be to find appropriate categorical definitions. I don't know how to define a local homeomorphism categorically, but maybe something similar to the story of separability (see [A. Carboni and G. Janelidze, Decidable (=separable) objects and morphisms in lextensive categories, Journal of Pure and Applied Algebra 110, 1996, 219-240] and [G. Janelidze and W. Tholen, Strongly separable morphisms in general categories, Theory and Applications of Categories 23, 5, 2010, 136-149]) can be done. It is also interesting that you say: "...Under hypotheses of the locally simply connected kind, unramified coverings are locally constant..." while what is done in my paper with Aurelio almost suggests to use the path lifting property to make a separable morphism a covering morphism (the occurrence of the path-lifting property is familiar of course, but the fact that it is "almost suggested" categorically a kind of new). All these problems, as well as the absence (so far!) of good Galois structures in TOP, Loc, Top, and Schemes, are related to each other, and more purely-categorical concepts are needed to understand these relationships better. Best wishes, George [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear George, Thank you for your comments and questions. I will have to look closely into the conditions of coverings in your sense to make sure that the off-my-hat assertion that unramified maps are an example is indeed correct. Also, your remark about the path-lifting property being "categorically forced" upon you in your work with Aurelio is particularly interesting to me as you suggested. I will most certainly look into those two points. I am also familiar with the T-categories and lax algebras, so that the work of Clementino, Hoffman and yourself should be easy for me to understand. I point out (in case you do not know this) that my paper "Coherent extensions and relational algebras", Trans. AMS 197 (1974) 355-390 introduces lax adjointness, examines examples including the T-categories of Burroni, and gives a new analysis of the example of topological spaces in this new light. I mentioned this paper to Maria Manuel Clementino the first time I heard her talk about this subject, as it was obviously relevant. However, in view of my imminent trip to Buenos Aires, where I will spend most of November, I doubt that I will find the time this week to look into all of that before I return. With best regards,Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill UniversityBurnside Hall, Office 1005 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810/3800 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/~bunge/ ************************************************ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (2)
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George Janelidze -
Marta Bunge