Recognizing mapping spaces

David Blanc
Tuesday, July 23, 2013 - from 14:00 to 14:40
Given a fixed object $A$ or $B$ in a suitable pointed simplicial model category $\C$, such as that of topological spaces, we study the problem of recovering $Y$ from the pointed mapping space \w{\mapa(A,Y)} or \w[.]{\mapa(Y,B)} In the first case, we describe a recognition principle, valid for any $A$, modelled on the classical ones for loop spaces, but using the more general notion of $A$-\emph{mapping algebra} \wh an enriched version of an algebra over a theory (or sketch). In the second case, our methods work in the more limited case when $B$ is an Eilenberg-Mac-Lane case (or similar inifinite loop space), providing a new perspective on the notion of $R$-completion. Joint work in parts with Hans-Joachim Baues, Bernard Badzioch and Wojciech Dorabiala, and Debasis Sen.