[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/ ]

Dear Bill, The history of simplicial terms is certainly complicated and often contradictory. Presently, the term of 'simplicial set/object' seems to be generally accepted (for a presheaf on positive finite ordinals) and this notion seems to be mostly considered as the leading one in its framework. My modest opinion is that simplicial terminology should be organised taking this term as the leading one.

It does not seem accurate to consider that there is an "opposition "between the ordered and unordered cases.

Fine, I agree, I should replace 'opposition' with 'distinction'. Certainly, I would not speak of opposition for the relationships set / preordered set space / directed space classical algebraic topology / directed algebraic topology. Your comments on distributive lattices are most interesting. Best wishes Marco On 24 Nov 2013, at 01:38, F. William Lawvere wrote:

Dear Marco If I understand the history, it is the term "simplicial sets" that was somewhat misleading : Classically the simplicial complexes (based ,from a categorical view, actually on finite families not just finite subsets) were an important combinatorial structure and indeed still are. Eilenberg and others recognized the important role in topological calculations for "ordered simplices " and the corresponding toposwas said to consist of "semisimplicial sets". But then the prefixes were dropped ! It does not seem accurate to consider that there is an "opposition "between the ordered and unordered cases. They are just two subtoposesof the distributive lattice classifier, which consists of presheaves onfinite posets. The classifying topos point of view is helpful, because it is the choice of an algebraic structure, of the kind classified, in a topological topos that gives rise to an exact singular/realization pair. For example any topological distributive lattice gives rise to such ;if it happens to be totally ordered, these adjoint functors factor throughthe simplicial subtopos. But a boolean algebra whose operations are continuous represents a realization that depends only on the subtoposthat depends only on the finite trivially ordered sets. The symmetry is concretely the action of boolean negation. The fact that any distributive lattice space embeds in a boolean algebra space helps to relate these . There are other subtoposes (i.e. strengthenings of the theory of distributive lattices), so that precise calculation of the Aufhebung andof coHeyting boundary should in particular be relevant to combinatorial topology. There may be a connection with your earlier work that employed distributive lattices in the analysis of diagrams. When one represents ageometric category in presheaves on a subcategory P, it is helpful to consider that P parameterizes the basic figure shapes that determine the structureof a general object. In the category of categories, the basic figures arefinite commutative diagrams, are they not ? Thus, if we do not insist on a minimalistic notion of nerve, distributive lattices emerge. Best wishesBill

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