Re: terminology for simplicial sets

[Peter May <may@math.uchicago.edu>]

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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