Recognizing mapping spaces

TL;DR

This paper develops a recognition principle for identifying the target object in a pointed simplicial model category from its mapping space, extending classical loop space techniques with a new ext{ m{A}}-space framework.

Contribution

It introduces a novel recognition principle based on ext{ m{A}}-spaces and a transfinite procedure for recovering objects from their mapping spaces.

Findings

Provides a new recognition criterion for mapping spaces.

Introduces a transfinite method for object recovery.

Extends classical loop space recognition techniques.

Abstract

Given a fixed object $A$ in a suitable pointed simplicial model category $\C$, we study the problem of recovering the target $Y$ from the pointed mapping space \w{\mapa(A,Y)} (up to $A$-equivalence). We describe a recognition principle, modelled on the classical ones for loop spaces, but using the more general notion of an \emph{\Ama[.]} It has an associated transfinite procedure for recovering \w{\CWA Y} from \w[,]{\mapa(A,Y)} inspired by Dror-Farjoun's construction of \ww{\CWA{}}-approximations.