Dear Peter, Ah, that is a condition I know well, thanks to work of Rina Foygel, a one-time student of mine now in our Statistics department. I had asked her to study the combinatorics of subdivision of categories. You can find a link to a talk that discusses the condition (starting on page 5) on my web page: http://www.math.uchicago.edu/~may Categories, posets, Alexandrov spaces, simplicial complexes, with emphasis on finite spaces. Buenos Aires, November 10, 2008 (dvi)(pdf) The property you ask about is called property A there, and using certain related properties B and C one can prove Theorem. A simplicial set K has property A if and only if its second barycentric subdivision Sd^2(K) is the simplicial set associated to a classical (ordered) simplicial complex. Another result is that if K does not have A, then Sd(K) cannot be a quasi-category. Still another is that if K has A, then Sd(K) is the nerve of a category. One transfers properties A, B, and C to categories via the nerve functor N. Using them, one proves Theorem. The second subdivision sd^2(C) of any category C is a poset. Theorem. For any category C, sd(C) is isomorphic to the `fundamental category' \tau_1(Sd(NC)). Theorem. A category C has property A if and only if Sd(NC) is isomorphic to N(sdC). Moreover, posets are equivalent to Alexandrov $T_0$-spaces, which have weakly homotopy equivalent classical simplicial complexes. These results, and others related to them, shed light on the Thomason model structure on Cat. Peter May On 10/19/10 5:54 AM, Prof. Peter Johnstone 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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