To Michael Batanin, for his nice questions 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. Contents 1. Homology with locally constant coefficients 1 2. Model structures for locally constant functors 6 3. Locally constant coefficients in Grothendieck derivators
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