Re: Undirected graph citation

[F W Lawvere <wlawvere@buffalo.edu>]

Dear George,

   Concerning undirected graphs, the Boolean algebra classifier, and
the intermediate sub-topos that suffices for groupoids:

   The special feature of these toposes I wanted to emphasize is not that
some of them can be generated by monoids, but rather (whether one splits
idempotents or not) that the site of operators is itself a full
subcategory of the category of sets. This is a small part of the point
that Vaughn wanted to make, I believe. Having the direct visualization of
this system of operators available as merely maps between certain small
sets is a useful auxiliary to formal presentations of the d,s kind.
As you mention, a basic way in which such presheaves can arise is by
applying a non-exact functor F to a group; the fact that the exponents on
the group are just these ordinary sets explains why we obtain an object
in this sort of topos (which can serve as a presentation of another group,
if desired).

   As noted in my unpublished (but widely distributed) paper on toposes
generated by codiscrete objects, the Yoneda embedding in these cases
produces of course a full sub-category of a topos, one which looks
exactly like (a piece of) the category of sets; of course this is not
the discrete inclusion, but its dialectical opposite, the codiscrete one.

Bill
************************************************************
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
************************************************************



On Fri, 3 Mar 2006, George Janelidze wrote:

> 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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