Re: preprint

[Marco Grandis <grandis43@hotmail.com>]

[Please do not use this e-address, but only the usual one at dima.unige.it]
Dear Bill,

Thank you very much for all these interesting comments and suggestions, we'll think of them.
We also received others from Ronnie Brown, Marta Bunge and Bob Pare; we  are preparing a revised version.

I am quite convinced of the importance of presheaves on nonempty finite sets. Just for the pleasure of friendly 
discussing, let me add that I like calling them 'symmetric simplicial sets' :-) - a term against which you protested 
sometime ago, in this list if I remember well; at least in a context where their links with simplicial sets and
algebraic topology are relevant.

On the other hand, I think that the classical term of 'simplicial complex' is misleading.
In fact, they sit inside symmetric simplicial sets (not inside simplicial  sets) as the objects that 
'live on their points', according to your terminology. I used for them the term 'combinatorial space',
while I used 'directed combinatorial space' for the (non classical) directed analogue (sitting inside 
simplicial sets), whose objects are defined by privileged finite sequences instead of privileged finite subsets.I think that the opposition between non-directed and directed notions should be kept clear.
Thanks again and best wishes    Marco

On 23/nov/2013, at 02.41, F. William Lawvere wrote:

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 of
discretes or constant presheaves. Because in this very special case
the 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 a
full 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 weak
infinite-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 of
essential 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 wishes
Bill

*the other example was the Gaeta topos on countably  infinite sets, 
closely related to the Kolmogoroff-Mackey bornological generalizations of
Banach spaces, which are  vector space objects in that topos. 		 	   		  

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