  The purpose of the present article is threefold. First of all, we rebuild the
whole theory of cosimplicial models of mapping spaces by using systematically
Kan adjunction techniques. Secondly, given two topological spaces X and Y, we
construct a cochain algebra which is quasi-isomorphic (as an algebra) to the
singular cochain algebra of the corresponding mapping space from X to Y. Here,
X has to be homotopy equivalent to the geometric realization a finite
simplicial set and of dimension less or equal to the connectivity of Y. At
last, we apply these results to the study of finite group actions on mapping
spaces that are induced by an action on the source.
