Dear Richard, Yes, that is relevant. But I don't think one can really use "semi-simplicial" as an adjective to qualify "simplicial set"; and I don't want to call them "semi-simplicial sets" since I want to think of them as simplicial sets with an additional property. I hesitate before adding one more mathematical usage of the word "regular"; but Myles Tierney informs me that it does at least have a respectable pedigree in this context (it was used by Steenrod), and it's surely better than Peter May's "Property A". Peter ---------------- On Wed, 20 Oct 2010, Richard Garner wrote:
Dear Peter,
Though I do not have a direct answer to your question the following seems at least relevant. By a semi-simplicial set, we mean a presheaf on Delta_f, the lluf subcategory of Delta spanned by the face operators (I think the terminology here is nowadays standard). The inclusion i: Delta_f -> Delta induces by left Kan extension a functor Lan_i from the category of semi-simplicial sets to the category of simplicial sets, which is faithful and, though not full, at least full on isomorphisms. Now a simplicial set satisfies the condition you name just when it lies in the replete image of this functor. This suggests that one might reasonably call a simplicial set satisfying your condition "semi-simplicial", and a map between two such "semi-simplicial" when it maps non-degenerate simplices to non-degenerate simplices. Another way of looking at it is that the semi-simplicial objects are those admitting coalgebra structure for the comonad (i^* o Lan_i) on simplicial sets; since Lan_i is full on isomorphisms, such structure will be unique up to unique isomorphism when it exists. The semi-simplicial maps between such objects are those which are coalgebra homomorphisms for some (and hence every) choice of coalgebra structure on their domain and codomain.
Richard
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