Enriched model categories and presheaf categories

Abstract

We collect in one place a variety of known and folklore results in enriched model category theory and add a few new twists. The central theme is a general procedure for constructing a Quillen adjunction, often a Quillen equivalence, between a given V-model category and a category of enriched presheaves in V, where V is any good enriching category. For example, we rederive the result of Schwede and Shipley that reasonable stable model categories are Quillen equivalent to presheaf categories of spectra (alias categories of module spectra) under more general hypotheses. The technical improvements and modifications of general model categorical results given here are applied to equivariant contexts in a pair of sequels, where we indicate various directions of application.

45 pages. v4. A number of relatively small changes and updates from the previous version, intended to address the most recent referee's report. The most significant change is the addition of section 4.5, which discusses Muro's work on arranging for a cofibrant unit

References in corpus (7)

• Higher Topos Theory
• Homotopy limits and colimits and enriched homotopy theory
• On the homotopy theory of enriched categories
• Models of G-spectra as presheaves of spectra
• Modules in Monoidal Model Categories
• Enriched model categories and an application to additive endomorphism spectra
• Constructing model categories with prescribed fibrant objects

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