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 On 19 October 2010 21:54, Prof. Peter Johnstone <P.T.Johnstone@dpmms.cam.ac.uk> wrote:
In something I've been thinking about recently, the condition on a simplicial set that all faces of non-degenerate simplices are non-degenerate seems to play a significant role. Does anyone know whether this condition has been considered previously, and if so whether it has a standard name?
The condition is of course satisfied by those simplicial sets which are derived from simplicial complexes in the standard way, but it's more general: it allows the possibility that two (formally) different faces of a non-degenerate simplex might coincide, as long as they're not degenerate.
Peter Johnstone
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