Re: property_vs_structure

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

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



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