John Baez and others interested in this question might begin by checking out the work of Rainer Vogt. Once when I was in Aarhus, he was telling me that a weak infinity category was a simplicial set that satisfied the Kan condition for interior faces only. That is, for any collection of n n-simplexes x^0,...,x^{i-1},x^{i+1},...x^n with 0 < i < n, such that d^jx^k = d^{k-1}x^j, whenever j < k and none of j, k, k-1 was i, there is an (n+1)-simplex x such that d^jx=x^j for j >< i. Applied in dimension 2, this gives a weak composite. In dimension 2 there is only one interior face and if this condition applied uniquely, you would have a category with d^1x being the composite of x^0 and x^2. Michael Barr ++++++++++++++++++++++++++++++++++++++++++