Subject: Question for list From: Allen Hatcher Date: Tue, 21 Dec 2004 22:38:19 -0500 To: Don Davis I have a two-part history/literature question about simplicial sets: 1. Where in the literature is there a statement of the elementary fact that a simplicial set is uniquely determined by its geometric realization (regarded as a CW complex whose cells have distinguished characteristic maps with domain a simplex)? Surely this fact is well known to the experts, but I haven't seen it in print. 2. Similarly, where is there a statement characterizing those CW complexes that arise as geometric realizations of simplicial sets? Roughly speaking, these are the CW complexes whose cells are attached by simplicial maps from the boundary of the standard simplex to the lower-dimensional skeleton, whose cells have simplicial structures by induction. A consequence of these elementary facts is that simplicial sets can be regarded as nothing more than CW complexes with a certain sort of extra simplicial structure on their cells. This viewpoint should be appealing to topologists with a penchant for geometry, although it of course runs counter to the whole idea of simplicial sets, which seems to be to replace geometry and topology in homotopy theory by combinatorics. For anyone who is interested, here is a link to a pdf file containing a couple pages describing this geometric interpretation of simplicial sets: http://www.math.cornell.edu/~hatcher/AT/ATsimplicial.pdf (This is intended to be a future addition to the Appendix of my Algebraic Topology book.) Allen Hatcher