As Clemens Berger reminds us, the category of small categories is a reflective subcategory of simplicial sets, with a reflector that preserves finite products. But as I mentioned, there is a similar "advantage" for the Boolean algebra classifier (=presheaves on non-empty finite cardinals, or "symmetric" simplicial sets): The category of small groupoids is reflective in this topos, with the reflector preserving finite products. Thus the Poincare' groupoid of a simplicial complex is directly available. (The simplicial complexes are merely the objects generated weakly by their points, a relation which defines a cartesian closed reflective subcategory of any topos.) It is not clear how one is to measure the loss or gain of combinatorial information in composing the various singular and realization functors between these different models. Is there such a measure? Bill Lawvere ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************