Re: property_vs_structure

[Marta Bunge <marta.bunge@mcgill.ca>]

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. 

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>> 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...



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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. 
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Best wishes,
Marta







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