Parametrized spaces model locally constant homotopy sheaves

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.

26 pages; exposition improved

References in corpus (3)

• Higher Topos Theory
• Homotopy limits and colimits and enriched homotopy theory
• Comparing composites of left and right derived functors

Cited by in corpus (7)

• Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory
• Units of ring spectra and Thom spectra
• Locally constant functors
• Comparing composites of left and right derived functors
• Morita cohomology
• Involutions of Azumaya algebras
• Generalized String Topology and Derived Koszul Duality