Re: property_vs_structure

["George Janelidze" <janelg@telkomsa.net>]

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.


...


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