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 Burnside 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/ ]
Dear Marta, I fully agree with every word you say, and I can only add: 1. After these many years I trust myself that covering morphisms should only be defined via Galois theory. But exactly for this reason any adjective should indicate that Galois theory is not applicable (or we do not know yet, how to apply it). So for me "unramified coverings" is one of possible good names for something that is not presented as coverings with respect to some Galois theory. 2. More importantly than terminology, I think to find what you call "generalized Galois theory" would be very interesting, and, as I already said many times, I should study your work seriously. And again, in my opinion the aim would be to find a general-categorical definition that gives good examples in all (or in the most of) those categories I mentioned before (that is, not just in Top and TOP, but also, say, in CR^o = the opposite category of commutative rings). Such investigations will - I believe - soon or late lead to a beautiful unification of certain big parts of algebraic topology and algebraic geometry. With best regards, George ----- Original Message ----- From: "Marta Bunge" <marta.bunge@mcgill.ca> To: <categories@mta.ca> Sent: Sunday, October 24, 2010 11:15 PM Subject: categories: Re: property_vs_structure 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. ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (2)
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George Janelidze -
Marta Bunge