Parametrized spaces model locally constant homotopy sheaves

TL;DR

This paper establishes a homotopy-theoretic framework connecting parametrized spaces, locally constant sheaves, and loop space actions, extending classical topological correspondences into a modern homotopical context.

Contribution

It proves that parametrized spaces fully embed into simplicial presheaves and characterizes their essential image as locally homotopically constant objects, extending classical covering space theory.

Findings

Homotopy theory of parametrized spaces embeds fully into simplicial presheaves.

Locally homotopically constant objects correspond to parametrized spaces.

Spaces over X are equivalent to spaces with loop space actions.

Abstract

We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopy-theoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that spaces parametrized over X are equivalent to spaces with an action of the loop space of X. This gives a homotopy-theoretic version of the correspondence between covering spaces over X and sets with an action of the fundamental group of X. We then use these two equivalences to study base change functors for parametrized spaces.