Mapping spaces and R-completion

We study the questions of how to recognize when a simplicial set X is of the form X= map _∗(Y, A) , for a given space A, and how to recover Y from X, if so. A full answer is provided when A= K(R, n) , for R= F_p or Q, in terms of a mapping algebra structure on X (defined in terms of product-preserving simplicial functors out of a certain simplicially enriched sketch Θ). In addition, when A= Ω ^∞A for a suitable connective ring spectrum A, we can recoverY from map _∗(Y, A) , given such a mapping algebra structure. This can be made more explicit when A= K(R, n) for some commutative ring R. Finally, our methods provide a new way of looking at the classical Bousfield–Kan R-completion.