Enriched model categories and presheaf categories

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.Comment: 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 uni

Models of G-spectra as presheaves of spectra

Let G be a finite group. We give Quillen equivalent models for the category of G-spectra as categories of spectrally enriched functors from explicitly described domain categories to nonequivariant spectra. Our preferred model is based on equivariant infinite loop space theory applied to elementary categorical data. It recasts equivariant stable homotopy theory in terms of point-set level categories of G-spans and nonequivariant spectra. We also give a more topologically grounded model based on equivariant Atiyah duality.Comment: 38 pages. v4. A number of relatively small changes and corrections. The introduction has been rewritten in response to suggestions from a referee. To make this paper more self-contained, section 4 on enriched model categories of G-spectra has been added. A new section 5 addresses a minor error in the previous version

Equivariant Iterated Loop Space Theory and Permutative \u3cem\u3eG\u3c/em\u3e–Categories

We set up operadic foundations for equivariant iterated loop space theory. We start by building up from a discussion of the approximation theorem and recognition principle for V–fold loop G–spaces to several avatars of a recognition principle for infinite loop G–spaces. We then explain what genuine permutative G–categories are and, more generally, what E∞–G–categories are, giving examples showing how they arise. As an application, we prove the equivariant Barratt–Priddy–Quillen theorem as a statement about genuine G–spectra and use it to give a new, categorical proof of the tom Dieck splitting theorem for suspension G–spectra. Other examples are geared towards equivariant algebraic K–theory