Re: property_vs_structure

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

Dear George,>

I haste to correct a possible misconception arising from my previous posting, and to propose an idea in connection with it. You wrote: 

>
> 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!

>
Let C = LoCo/E, defined as in your book Galois Theories (with F. Borceux). An object p  of C with domain F is said to be a covering morphism if there exists a morphism e of effective descent in Top with codomain E such that (F,p) is split by e.  A complete spread p of C with domain F  - that is, an unramified morphism, need not be a covering morphism in C  in your sense, as we know. Whether there is a Galois structure on C whose coverings are precisely the unramified coverings without it forcing them to be identified with the locally constant coverings does not seem likely. At least we know that the class of unramified coverings in C is stable under pullbacks and has other nice properties, so C is a natural choice of universe. The fact that we have called "coverings" the unramified morphisms may then be misleading if coverings are to be tied up with Galois theory. 

>
Nevertheless, it is the case in topology that the notion of a covering in  the traditional sense has been enlarged to include branchings but not folds. A branched covering of a locally connected space E (R.H. Fox 1957)  is  the spread completion of a locally constant covering on a pure open subspace U of E, thought off as "the complement of a knot". At least in this case it is meaningful to consider the "branched fundamental groupoid" (or "knot groupoid") of E with non-singular part U. It might be of interest to consider a notion of  "generalized Galois theory" to encompass this notion of "generalized covering morphism". Let me know what you think. Relevant discussions in topos theory can be found in (Bunge-Niefield 2000), (Funk 2000),  (Bunge-Lack 2003), as well as in (Bunge-Funk 2006, 2007). 

>

With best regards,

>

Marta




************************************************
Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics
McGill University
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Montreal, QC, Canada H3A 2K6
Office: (514) 398-3810/3800
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marta.bunge@mcgill.ca
http://www.math.mcgill.ca/~bunge/
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