In his letter of 2 August, Phillippe Gaucher asks for a characterization of those simplicial sets that are lambda-presentable for a regular cardinal lambda. One might as well take lambda to be aleph_0, and seek the finitely presentables, the passage from this case to that of a general lamvda being trivial. The simplicial sets form the functor-category [K,Set] where K is (DELTA)^op. For any small K, the representables in A = [K,Set] form a dense full subcategory consisting (by the Yoneda lemma) of finitely presentable objects of A. Accordingly A is locally finitely presentable, and the finitely-presentables of A form the closure in A of the representables under finite colimits. Clearly this consists of those F : K --> Set such that Fk is finite for each object k of K, and is empty for all k except those in a finite subset of K.
The last condition is unclear to me: it is not satisfied by the representables themselves. I believe finite colimits of representables in this case are those simplicial sets which have only finitely many *nondegenerate* simplices. (A simplex Delta^k -> F is nondegenerate iff it is mono; it so happens that here any quotient of a single representable is representable again).