The following preprint is available: M. Grandis Simplicial toposes and combinatorial homotopy, Dip. Mat. Univ. Genova, Preprint 400 (1999). Abstract. The term *combinatorial topos* denotes here a topos of presheaves over a small subcategory of the category of finite sets. The main instances we want to consider are the presheaf categories of simplicial sets, cubical sets, and globular sets, together with their symmetric versions: e.g., the topos !Smp of symmetric simplicial sets consists of all presheaves on the category !Delta of finite, positive cardinals. We show here how combinatorial homotopy, developed in previous works for simplicial complexes (the cartesian closed subcategory of *simple* presheaves in !Smp) can be extended to the topos !Smp. As a crucial advantage, the (extended) fundamental groupoid Pi_1: !Smp --> Gpd is left adjoint to a natural functor M_1: Gpd --> !Smp, the symmetric nerve of a groupoid, and therefore - as a strong van Kampen property - preserves all colimits. Analogously, a notion of (non-reversible) *directed* homotopy can be developed in Smp, with applications to image analysis similar to the ones of the symmetric case. We have now a homotopy n-category functor C_n: Smp --> n-Cat, left adjoint to a nerve N_n = n-Cat(C_n(Delta[n]), -). It would be interesting to determine whether the n-category C_n(Delta[n]) coincides with Street's oriental O_n, and the previous nerve with Street's, as it seems likely. ___ Available at: ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/CmbTop.Sep99.ps (459 K) ___ Marco Grandis Dipartimento di Matematica Universita' di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it tel: +39.010.353 6805 fax: +39.010.353 6752 http://www.dima.unige.it/STAFF/GRANDIS/ ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/