Re: preprint

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

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
> From: grandis43@hotmail.com
> To: wlawvere@hotmail.com; categories@mta.ca
> Subject: categories: Re: preprint
> Date: Sat, 23 Nov 2013 12:03:56 +0100
> 
> [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
> 

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