Locally constant functors

TL;DR

This paper explores the theory of locally constant functors and their role in homotopy theory, using homotopy Kan extensions, Bousfield localization, and derivators to deepen understanding of small categories.

Contribution

It introduces a comprehensive framework connecting locally constant functors with homotopy Kan extensions, Bousfield localization, and derivators, advancing the theoretical understanding of homotopy theory of categories.

Findings

Homotopy Kan extensions characterize the homotopy theory of small categories.

Bousfield localization of diagram categories provides a model for locally constant functors.

Derivators offer a unifying perspective on homotopy Kan extensions and locally constant functors.

Abstract

We study locally constant coefficients. We first study the theory of homotopy Kan extensions with locally constant coefficients in model categories, and explain how it characterizes the homotopy theory of small categories. We explain how to interpret this in terms of left Bousfield localization of categories of diagrams with values in a combinatorial model category. At last, we explain how the theory of homotopy Kan extensions in derivators can be used to understand locally constant functors.