Homotopy Theory of Enriched Mackey Functors

TL;DR

This paper develops a framework for enriched Mackey functors over closed multicategories, exploring how changing enrichments via K-theory multifunctors affects their homotopy theory, with applications to spectra and permutative categories.

Contribution

It introduces a detailed theory of enriched Mackey functors over multicategories and analyzes the effects of enrichment changes on their homotopy categories.

Findings

Equivalences of homotopy theories induced by K-theory multifunctors.

Framework for diagrams and Mackey functors enriched in multicategories.

Applications to spectra, permutative categories, and pointed multicategories.

Abstract

Mackey functors provide the coefficient systems for equivariant cohomology theories. More generally, enriched presheaf categories provide a classification and organization for many stable model categories of interest. Changing enrichments along $K$-theory multifunctors provides an important tool for constructing spectral Mackey functors from Mackey functors enriched in algebraic structures such as permutative categories. This work gives a detailed development of diagrams, presheaves, and Mackey functors enriched over closed multicategories. Change of enrichment, including the relevant compositionality, is treated with care. This framework is applied to the homotopy theory of enriched diagram and Mackey functor categories, including equivalences of homotopy theories induced by $K$-theory multifunctors. Particular applications of interest include diagrams and Mackey functors enriched in pointed multicategories, permutative categories, and symmetric spectra.