Re: Undirected graph citation

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

I am not sure if I really understand what is the target of this discussion,
but I would like to make some comments to Bill's messages:

The Poincare' groupoid is (up to an equivalence) nothing but the largest
Galois groupoid, and it is directly available as soon as one has what I call
Galois structure in my several papers, if we assume that every object of the
ground category has a universal covering. This is certainly the case for
every locally connected topos with coproducts and enough projectives.
Therefore this is certainly the case for every presheaf topos. Therefore
what Bill means by "directly available" should be not "available without
going through geometric realization" but just "can be calculated as the
result of reflection" (probably this is exactly what Bill had in mind).

Moreover, it was Grothendieck's observation that Galois/fundamental
groupoids are to be defined as quotients of certain equivalence relations -
in fact kernel pairs, and this observation was used by many authors in topos
theory and elsewhere; my own observation (1984) then was that one can make
Galois theory purely categorical by using not "quotients" but "images under
a left adjoint" (the first prototype for me was actually not Grothendieck's
but Andy Magid's "componentially locally strongly separable" Galois theory
of commutative rings). What I am trying to conclude is that the
Galois/fundamental groupoids actually arise not from anything simplicial but
from abstract category theory: it is just a result of a game with adjoint
functors between categories with pullbacks.

In another message Bill says: "A similar lacuna of explicitness occurs in
many papers on Galois theory where pregroupoids are an intermediate step ;
the description of  the pregroupoid concept is really just a presentation
of the monoid of endomaps of the 4-element set..." Assuming that everyone
understands that this is not about classical Galois theory (I don't think
somebody like J.-P. Serre ever mentions pregroupoids) and not about what
Anders Kock calls pregroupoids, let me again return to the categorical
Galois theory:

If p : E ---> B is an "extension" in a category C, R its kernel pair, and F
: C ---> X the left adjoint involved in a given Galois theory, then one
wants to define the Galois groupoid Gal(E,p) as F(R) = the image of R under
F (I usually write I instead of F, but in an email message this does not
look good...). But if our extension p : E ---> B is not normal, then, since
F usually does not preserve pullbacks, F(R) is not a groupoid - it is a
weaker structure, the "equational part" of groupoid structure. This weaker
structure is still good enough to define its internal actions in X and these
internal actions classify covering objects over B split by (E,p). Hence this
weaker structure needs a name and I called it "pregroupoid" (I did not know
that this term was already overused for almost the same and for unrelated
concepts). I cannot speak for everyone, but for my own purposes there are
actually several possible candidates for the notion of pregroupoid and half
of them can certainly be defined as monoid actions for a specific monoid,
like the one Bill mentions. However, in each case we deal with a "very
small" category actions and it is a triviality to observe that that category
can be replaced with a monoid. Essentially, what you need is to check that
the terminal object (in your category of pregroupoids) has either no proper
subobjects or only one such, which must be initial. In this observation -
due to Max Kelly, about the categories monadic over powers of Sets being
monadic over Sets, one usually says "strictly initial"; but we can omit
"strictly" here since it is about a topos.

George Janelidze

----- Original Message -----
From: "F W Lawvere" <wlawvere@buffalo.edu>
To: <categories@mta.ca>
Sent: Thursday, March 02, 2006 8:32 PM
Subject: categories: Re: Undirected graph citation


>
> As Clemens Berger reminds us, the category of small categories
> is a reflective subcategory of simplicial sets, with a reflector that
> preserves finite products. But as I mentioned, there is a similar
> "advantage" for the Boolean algebra classifier (=presheaves on non-empty
> finite cardinals, or "symmetric" simplicial sets):
> The category of small groupoids is reflective in this topos, with the
> reflector preserving finite products. Thus the Poincare' groupoid of a
> simplicial complex is directly available. (The simplicial complexes are
> merely the objects generated weakly by their points, a relation which
> defines a cartesian closed reflective subcategory of any topos.)
>
> It is not clear how one is to measure the loss or gain of combinatorial
> information in composing the various singular and realization functors
> between these different models. Is there such a measure?
>
>
> Bill Lawvere
>
> ************************************************************
> F. William Lawvere
> Mathematics Department, State University of New York
> 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA
> Tel. 716-645-6284
> HOMEPAGE:  http://www.acsu.buffalo.edu/~wlawvere
> ************************************************************
>
>
>
>
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