Re: terminology for simplicial sets

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

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
suggested semisimplicial, but that to my mind is certainly not better
(but then I'm old enough to remember when semisimplicial meant
what we now call simplicial, to differentiate from classical simplicial
complexes).

Peter


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