[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 morphism 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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]