# Detecting Mapping Spaces

## TL;DR

This paper characterizes finite CW-complexes for which algebraic theories can detect mapping spaces, showing they must have the homology of a wedge of spheres, and in the simply connected case, they are homotopy equivalent to such wedges.

## Contribution

It establishes a precise homological and homotopical classification of spaces detectable by algebraic theories in the context of mapping spaces.

## Key findings

- Spaces detected by algebraic theories have the homology of a wedge of spheres.
- Simply connected such spaces are homotopy equivalent to wedges of spheres.
- Provides conditions under which algebraic theories detect mapping spaces.

## Abstract

We show if $A$ is a finite CW-complex such that algebraic theories detect mapping spaces out of $A$, then $A$ has the homology type of a wedge of spheres of the same dimension. Furthermore, if $A$ is simply connected then $A$ has the homotopy type of a wedge of spheres.