22 Oct 03:43 2010

## Re: terminology for simplicial sets

Peter May <may <at> math.uchicago.edu>

2010-10-22 01:43:08 GMT

2010-10-22 01:43:08 GMT

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(Continue reading)