22 Oct 03:43 2010

Re: terminology for simplicial sets

I agree with Peter J. that a good name is a good thing.

However, he has suggested that regular'' is surely better than
Property A'', and I have to say that it most assuredly is not.
Regular CW complex has a standard meaning, and there is a
related standard meaning for regular simplicial set, which is
recalled on page 4 of the link I sent originally
(http://www.math.uchicago.edu/~may
Categories, posets, Alexandrov spaces, simplicial complexes,
with emphasis on finite spaces. Buenos Aires, November 10, 2008)

A nondegenerate  n-simplex x in a simplicial set K is regular if the
subcomplex [x] that it generates is the pushout of  the last face
inclusion  \Delta_{n-1}  \to  \Delta_{n}  along  the last face
d_n x : \Delta[n-1] \to [d_nx].   K itself is regular if all of its
nondegenerate simplices are regular.

The subdivision of any simplicial set is regular.

This definition is standard because  the realization of a regular
simplicial
set is a regular CW complex, and regular CW complexes are triangulable,
that is homeomorphic to the realization of a simplicial set coming from a
classical simplicial complex.  This is all classical, and I could cite a
number
of sources. A modern one (1990) with a good treatment is Fritsch and
Piccinini,  Cellular structures in topology.  See p. 208.

I hold no particular brief for Property A (and B and C), but they
will do until something definitely better comes along. Richard has