I have no opinion on the notational question. I suppose the standard notation is backward, but I am not going to change now. I use d^0 and d^1, BTW, reserving the lower index for the dimension, which in this case is 1. If an equational category has a Mal'cev operator (a ternary operation t with t(x,x,y) = t(y,x,x) = y), then every simplicial object is Kan. This is essentially clear from John Moore's proof of the group case, in which forms xy^{-1}z appear repeatedly. For an equational category, the converse is also true, since a special case of every simplicial object being Kan is that every reflexive relation is an equivalence relation, a well-known characterization of Mal'cev. Most familiar Mal'cev categories have a group structure, so the Kan condition follows from the case for groups. Just about the only familiar Mal'cev category that I can think of that lacks a group op is Heyting algebras and I don't know any application of simplicial Heyting algebras. --Michael Barr