## Abstract

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.

Original language | English |
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Pages (from-to) | 635-671 |

Number of pages | 37 |

Journal | Journal of Homotopy and Related Structures |

Volume | 13 |

Issue number | 3 |

DOIs | |

State | Published - 1 Sep 2018 |

### Bibliographical note

Publisher Copyright:© 2018, Tbilisi Centre for Mathematical Sciences.

## Keywords

- Cosimplicial resolution
- Mapping algebra
- Mapping space
- Rationalization
- p-Completion

## ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology