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= Fp 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.
Bibliographical noteFunding Information:
Acknowledgements We wish to thank Pete Bousfield for many useful comments and elucidations of his work, and Paul Goerss for a helpful pointer. We also thank the referee for his or her comprehensive and helpful suggestions. The research of the first author was partially supported by Israel Science Foundation Grants 74/11 and 770/16, and that of the second author by INSPIRE Grant IFA MA-12.
© 2018, Tbilisi Centre for Mathematical Sciences.
- Cosimplicial resolution
- Mapping algebra
- Mapping space
ASJC Scopus subject areas
- Algebra and Number Theory
- Geometry and Topology