Last night I wrote a silly response to Phillippe Gaucher's question of 2 July, asking for the lambda-presentable simplicial sets, where lambda is a regular cardinal. By way of excuse, it was 3am here, and I had turned on the email when I had trouble sleeping. As I said, the case lambda = aleph_0 of the finitely-presentables illustrates the case of a general lambda. The simplicial sets form the presheaf category A = [K,Set] wkere K is (DELTA)^op. Each representable K(k,-) is finitely presentable (by the Yoneda lemma), and the representables are dense in A, so that A is locally finitely presentable, its finitely presentables constituting the closutr in A under finite colimits of the representables. Thus far I was correct last night, but I was confused about the nature of the representables; the answer I gave was correct for the case of a discrete category K. When K = (DELTA)^op, the representables are the simplexes, and even the 0-simplex has a non-empty component in each dimension. In the case of a presheaf category [K,Set], things are simpler than in a more general locally finitely presentable category. The closure under finite colimits of the representables consists of the finite colimits of the representables; and this in turn consists of the coequalizers of all the diagrams f,g: P --> Q where P and Q are finite coproducts of representables (that is, of simplexes in the simpllicial sets case). See Satz 7.6 of [P.Gabriel and F.Ulmer, Lokal praesentierbare Kategorien, Springer Lecture Notes in Math.221 (1971)]. Max Kelly.