Parametrized spaces model locally constant homotopy sheaves

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.

💡 Research Summary

The paper establishes a precise homotopy‑theoretic equivalence between parametrized spaces (spaces equipped with a map to a fixed base space) and locally constant homotopy sheaves (simplicial presheaves that are homotopically constant on each open subset of the base). The authors begin by recalling two classical viewpoints: (i) sheaves (or stacks) of spaces on a base, and (ii) spaces over a base (parametrized spaces). While both approaches have been used in various contexts—such as parametrized cohomology theories—the literature lacks a direct comparison of their homotopy theories.

To bridge this gap, the authors first adopt a mixed model structure on the category of topological spaces, originally due to Cole. This “m‑model” combines the standard Quillen model (weak homotopy equivalences, Serre fibrations) with the classical Hurewicz model (homotopy equivalences, Hurewicz fibrations). In this mixed setting, weak equivalences are the usual weak homotopy equivalences, fibrations are Hurewicz fibrations, and cofibrations are Hurewicz cofibrations that factor as a Quillen cofibration followed by a homotopy equivalence. Crucially, the m‑cofibrant objects are precisely those having the homotopy type of CW complexes, which allows the authors to control local finiteness and compactness properties needed later.

Next, the paper equips the category of simplicial presheaves on a base space (B) with the Bousfield–Kan model structure, where weak equivalences are stalkwise weak equivalences. This yields a homotopy theory of “homotopy sheaves.” The authors then construct a Quillen functor \