A note on localizations of mapping spaces

TL;DR

This paper investigates the conditions under which localization functors preserve mapping spaces, revealing that such preservation occurs only when the domain has a specific rational homotopy type.

Contribution

It establishes a precise criterion linking the preservation of mapping spaces by localization to the rational homotopy type of the domain complex.

Findings

Localization preserves mapping spaces only for certain rational homotopy types.

The domain must have the rational homotopy type of a wedge of spheres of a fixed dimension.

Provides a characterization of when localization functors preserve mapping spaces.

Abstract

We show that if A is a simply connected, finite, pointed CW-complex then the mapping spaces Map(A, -) are preserved by the localization functors only if A has the rational homotopy type of a wedge of spheres of a fixed dimension.