Re: property_vs_structure

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

Dear All,

Apologizing to those who heard this many times, I would like to say - since
we are talking about "property_vs_structure" for covering maps:

What I called "Galois structures" more than 20 years ago were exactly the
structures needed to define covering morphisms in general categories. It is
well known (although it is not clear what it really means!) that Poincare's
first ideas about covering maps were inspired by Galois theory, and so
algebra was "there" even before topology. Having also in mind Grothendieck's
work, topos-theoretic developments, and Magid's work, I am certainly not
original in saying that covering maps should belong to category theory
rather than to topology.

A Galois structure (say, on a category C) essentially consists of three
ingredients (although various modifications are possible):

(i) A functor I : C ---> X. It should better have a right adjoint, and it  is
wonderful if it is semi-left-exact or, which is almost the same in a sense,
if it is a fibration. But relative versions of these conditions involving  F
below are also good.

(ii) A class F of morphisms in C. All covering morphisms we are going to
define will be inside F.

(iii) Another class E of morphisms in C. Although usually I do not mention
it separately because I prefer to define it as the class of effective
F-descent morphisms. According to the terminology, Marta and Eduardo used  in
their messages, E should now be called a "trivialization structure".

For a given Galois structure, the covering morphisms are defined as in my
papers, and I would repeat the question about "property vs structure" for
covering morphisms as follows:

Given a category C, where the concept of a covering seems to be important,
do we want to fix "the best" Galois structure, or we should consider
several/many such structures?

It is certainly a matter of taste, but to feel the taste one surely needs
examples. In my opinion the ones listed below are especially important; they
were investigated together with several people you know, whom I was very
honoured to work with.

Example 1. C = the opposite category of commutative rings. Here we have  a
very good candidate, which is: X = the category of Stone spaces; I : C --->
X = the Boolean spectrum functor; F = the class of all morphisms in C; E =
the class of all effective descent morphisms in C. In this case covering
morphisms are the same what A. R. Magid called componentially locally
strongly separable algebras (considering an R-algebra A as a morphism A --->
R in C), and are THE most general algebras for which he developed his
"separable Galois theory" presented in [A. R. Magid, The separable Galois
theory of commutative rings, Marcel Dekker, 1974].

Example 2. C = the category of locally connected topological spaces. Here
again, we seem to have "the best candidate". It is: X = the category of
sets; I : C ---> X = the functor sending spaces to the sets of their
connected components; F = the class of local homeomorphisms of locally
connected spaces; F = the class of surjective maps from E. The covering
morphisms are then the same the covering morphisms of locally connected
spaces in the usual sense.

Example 3. C is a locally connected topos. This essentially generalizes the
previous example and everything happens as there (although here F is the
class of all morphisms and E the class of all epimorphisms in C of course).
Marta knows much more than I do about this example and its connections with
other topos-theoretic constructions; my only contribution is the short paper
[G. Janelidze, A note on Barr-Diaconescu covering theory, Contemporary
Mathematics 131, 3, 1992, 121-124].

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...

Example 5. C = the category of compact Hausdorff spaces. Here "the best
candidate" seems to be: X = the category of Stone spaces; I : C ---> X
sending compact Hausdorff spaces to the Stone spaces of their connected
components; F = the class of all morphisms in C; E = the class of all
morphisms in C that are surjections. As shown in [A. Carboni, G. Janelidze,
G. M. Kelly, and R. Paré, On localization and stabilization of factorization
systems, Applied Categorical Structures 5, 1997, 1-58], the covering
morphisms here are the same as light maps in the sense of Eilenberg and
Whyburn.

Example 6. C = the category of simplicial sets. Since it is a category of
the form Fam(A), Example 4 can be used. However, [R. Brown and G. Janelidze,
Galois theory of second order covering maps of simplicial sets, Journal of
Pure and Applied Algebra 135, 1999, 23-31] gives a Galois structure that
produces a larger (new) class of covering morphisms. In that structure X is
the category of groupoids; I : C ---> X the fundamental groupoid functor,  F
the class of Kan fibrations, and E the class of surjective Kan fibrations.
Another "simplicial Galois theory" is presented in [M. Grandis and G.
Janelidze, Galois theory of simplicial complexes, Topology and its
Applications 132, 3, 2003, 281-289].

Example 7. C = the category of groups. The "most classical" candidate would
be: X = the category of abelian groups; I : C ---> X = the abelianization
functor; E = F = the class of group epimorphisms. In this case the covering
morphisms are the same as central extensions. This was my first example of a
"very-non-Grothendieck Galois theory". A bit later I realized that C can be
replaced with any variety of groups with multiple operators in the sense of
[P. J. Higgins, Groups with multiple operators, Proc. London Math. Soc.
(3)6, 1956, 366-416] and X with any subvariety in C - and then we get
central extensions relative to a subvariety in the sense of A. Fröhlich's
school (see e. g. [A. Fröhlich, Baer-invariants of algebras, Trans. AMS  109,
1963, 221-244], [A. S.-T. Lue, Baer-invariants and extensions relative to  a
variety, Proc. Cambridge Philos. Soc. 63, 1967, 569-578], [J.
Furtado-Coelho, Varieties of W-groups and associated functors, Ph.D. Thesis,
University of London, 1972]). The next step was to get rid of groups
completely and Max Kelly and I found out that the crucial property that
helps to work with generalized central extensions is congruence modularity,
and we wrote [G. Janelidze and G. M. Kelly, Galois theory and a general
notion of a central extension, Journal of Pure and Applied Algebra 97, 1994,
135-161]. Many further results were obtained by Marino Gran, partly in
collaboration with Dominique Bourn (see [M. Gran, Applications of
categorical Galois theory in universal algebra, Fields Institute
Communications 43, 2004, 243-280] and references there for what was done
until 2002/3), Tomas Everaert and Tim Van der Linden, and by Tim and Tomas
separately and together. Writing this I feel now bad not to say more about
their brilliant results, also involving higher central extensions (see in
particular [T. Everaert, M. Gran, and T. Van der Linden, Higher Hopf
formulae for homology via Galois theory, Advances in Mathematics 217, 2008,
2231-2267]). I shall gladly say more at another occasion.

Examples 8-11?. Let C be one of the following three categories: (a)
topological spaces; (b) locales; (c) toposes; (d) schemes of algebraic
geometry (although (c) is 2-dimensional). I still do not know anything like
"the best candidates"...

And finally there are trivial examples: (a) C = X, with I the identity
functor; and (b) X = 1. Taking (in both of them) E to be the class of all
morphisms in C (and suitable F) we will have: all morphisms in C are
covering morphisms in the situation (a), and only isomorphisms are covering
morphisms in the situation (b).

In particular, since "the largest" Galois structure is trivial, I would
conclude: there is no "the best" Galois structure, and one should rather
consider several/many such structures.

George



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