RE: preprint

["F. William Lawvere" <wlawvere@hotmail.com>]

Dear Ettore and Marco
Much of your very interesting paper is actually taking place in an unjustly  neglected topos.
The classifying topos for nontrivial Boolean algebras is concretely just presheaves on nonempty finite sets. It was one of two examples in my 1988 Macquarie talk "Toposes generated by codiscrete objects in 
algebraic topology and functional analysis" *
Like many non-localic toposes, this one unites pair of identical  but adjointly opposite copies of the base topos of abstract sets,namely a subtopos of codiscretes and its negation, the subcategory ofdiscretes or constant presheaves. Because in this very special casethe Yoneda inclusion is a restriction of  the codiscrete inclusion, it is well justified to consider  this topos as "codiscretely generated " (wrt colimits).
This topos contains the category of groupoids as a full reflective subcategory.In fact the reflector preserves finite products, and is a refinement of the syntax for presenting groups. It should probably be considered as the fundamental combinatorial Poincare functor .
That point of view requires viewing the objects of the topos as combinatorial spaces, which  is out of fashion even though afull coreflective subcategory is that of classical simplicial complexes ( or "simplicial schemes"according to Godement).  The fear is that geometric realization will not be exact. But as Joyal and Wraith pointed out 30 years ago, one need only replace the usual interval  with the weakinfinite-dimensional sphere as basic parameterizer (of Zitterbewegungen instead of ordinary paths ?) in order to  obtain  a geometric mo\x10rphism of the singular/realization kind from any reasonable topological topos.
Actually the groupoids  are already at a low level in the sequence ofessential subtoposes: if the trivial topos is taken as level minus infinity,the  discrete as level zero, and the lowest level whose skeleton functor detects connectivity as level one, then the lowest level whose UIAO is compatible seems to be three.  Explicit consideration of the role of the groupoids may a  help in calculating the precise Aufhebung  function.
Best wishesBill
*the other example was the Gaeta topos on countably  infinite sets, closely related to the Kolmogoroff-Mackey bornological generalizations ofBanach spaces, which are  vector space objects in that topos.






> From: grandis@dima.unige.it
> Subject: categories: preprint
> Date: Mon, 18 Nov 2013 11:51:29 +0100
> To: categories@mta.ca
> 
> The following preprint is available:
> 
> -------
> E. Carletti - M. Grandis
> Fundamental groupoids as generalised pushouts of codiscrete groupoids
> Dip. Mat. Univ. Genova, Preprint 603 (2013).
>  	http://www.dima.unige.it/~grandis/GpdClm.pdf
> 
> Abstract. Every differentiable manifold X has a ‘good cover’, where   
> all open sets and their finite intersections are contractible. Using  
> a generalised van Kampen theorem for open covers we deduce that the  
> fundamental groupoid of X is a ‘generalised pushout’ of codiscrete  
> groupoids and inclusions.
> This fact motivates the present brief study of generalised pushouts.  
> In particular, we show that every groupoid is up to equivalence a  
> generalised pushout of codiscrete subgroupoids, and that (in any  
> category) finite generalised pushouts amount to ordinary pushouts and  
> coequalisers.
> -------
> 
> Before submitting it, I would like to know if the ‘generalised  
> pushouts’ we are using (or similar colimits) have been considered  
> elsewhere.
> (They are not simply connected colimits, in the sense of Bob Pare,  
> and indeed they cannot be constructed with pushouts.)
> 
> With best regards to all colleagues and friends. In particular to  
> Ronnie Brown and Bob Pare, whose results are used in this preprint.
> 
> Marco
> 

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