Bonjour, I think that I have found a small pathological behaviour with dimension of morphisms in omega-categories. This is the following one. Let us first recall that min{m/s_m(x)=x} = min{n/t_n(x)=x} =: dim(x). Because x = s_m(x) = t_m s_m(x) (because m=<m) = t_m(x). Lemma 1 : Let u and v be two composable morphisms of dimension 1 (i.e. t_0(u)=s_0(v)). Suppose that t_0(v)<>s_0(u) (<> means different). Then u o_0 v is of dimension 1. Proof : Suppose that u o_0 v of dimension 0. Then s_0(u) = s_0(u o_0 v) = u o_0 v and t_0(v) = t_0(u o_0 v) = u o_0 v So s_0(u) = t_0(v) : false. Here is now the pathological behaviour : If s_0(u) = t_0(v), here is an example of category with u and v of dimension 1 and u o_0 v of dimension 0 : We take A={alpha,beta,u,v} (four elements) ------------------------------------- alpha beta u v ------------------------------------- s_0 alpha beta alpha beta ------------------------------------- t_0 alpha beta beta alpha ------------------------------------- and for o_0 : ------------------------------------- alpha beta u v ------------------------------------- alpha alpha xxxx u xxxx ------------------------------------- beta xxxx beta xxxx v ------------------------------------- u xxxx u xxxx alpha ------------------------------------- v v xxxx beta xxxx ------------------------------------- (at the intersection of row x and column y, read x o_0 y) All axioms of categories seem to be verified (?). The only composition is a "word" of finite length "... u (beta) v (alpha) u (beta) v..." In (A,s_0,t_0,o_0), u and v are of dimension 1 and u o_0 v of dimension 0 because u o_0 v = alpha. Where am I wrong ? (observe that this behaviour cannot appear with for example a category coming from a composable pasting scheme). pg.