Re: fundamental_localic_groupoid?

["Joyal, André" <joyal.andre@uqam.ca>]

David Roberts wrote:


>Going back to more general thoughts, given an arbitrary space X, the topos
>Sh(X) is the classifying topos of some localic groupoid G_X (Joyal-Tierney),
>which I thought (!) could be interpreted as the fundamental groupoid in nice
>situations. Here I suppose is my real question: what relation is there
>between G_X and the (ordinary) fundamental groupoid Pi_1(X)? I suppose a
>test case could be the Warsaw circle W: Pi_1(W) is equivalent to the
>discrete groupoid on the set {a,b}, and surely G_W is more than this.

I would like to make a few observations.
If my recollection is right,  during the 1980's the dream of some topos theorists 
(including myself) was to use atomic toposes as generalised K(pi,1)-spaces.
The dream has not materialised and it maybe foolish.

Another avenue is to extend Grothendieck Galois theory, since the
fundamental groupoid of a space classifies the covers of that space.
Eduardo Dubuc is presently developing a general theory of Galois toposes.
The fundamental Galois topos of a topos E could be defined as the 
reflection E--->E(1) of E into the full subcategory of Galois toposes
over the base topos S. I am not supposing that S is the category of sets.
I am writing E(1) because I want to suggest that it is a stage of
a Postnikov tower of S-toposes: 

E(-1)<---E(0)<---E(1)<----E(2)<----

The topos E(-1) is of course the image of the 
geometric morphisme E-->S. The construction of E(-1) 
depends on the factorisation system 

(epimorphisms of toposes, sub-toposes)

The topos E(0) is the totally disconnected localic reflection of E
over the base topos S. Its construction
depends on the factorisation system 

(connected morphisms of toposes, totally disconnected localic morphisms)

The topos E(1) is the Galois topos reflection of E 
over the base topos S. Its construction
depends on the factorisation system 

(1-connected morphisms of toposes, Galois morphisms)


The topos E(2) does not exists in general for the
simple reason that it is a 2-topos, not an ordinary
topos. A 2-topos is enriched over groupoids.
 
The whole theory depends on three factorisation systems 

(epimorphisms of toposes, sub-toposes)
(connected morphisms of toposes, totally disconnected localic morphisms)
(1-connected morphisms of toposes, Galois morphisms)

The first factorisation system is well known and 
was constructed a long time ago.

Excuse my ignorance, but I do not know if the 
second factorisation system has been fully constructed. 
Can someone tell me?

The third factorisation system was partially
constructed by Dubuc: he can factor
the morphism E-->S when S is the category of sets.

My observations concerning your problems
could be off the mark.


Best,
André




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