Mapping spaces and R-completion

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