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
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