Dear Peter, Many thanks. Naturally, I'd been hoping that it might have a more descriptive name than "Property A", but if that is what it's called ... The reason I got interested in it: if you consider the total category of the discrete fibration (over the simplicial category Delta) corresponding to a given simplicial set, the full subcategory whose objects are the non-degenerate simplices is reflective iff Property A holds. Peter On Tue, 19 Oct 2010, Peter May wrote:
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
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