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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]