This chapter reviews the use of enriched diagrams and enriched Mackey functors in equivariant homotopy theory and the theory of stable model categories. The results outlined here are not used directly in the main body of the work, but they provide an important motivating context. The goal of this chapter is, therefore, to outline some of the main ideas and provide numerous references to the literature for further treatment.

This chapter provides the main results of Part 3. These make use of the preceding material on enrichment over (closed) multicategories, and apply it to categories of enriched diagrams and enriched Mackey functors. A key detail, both here and in the homotopical applications of Part 4, is that nonsymmetric multifunctors provide a diagram change of enrichment, but not necessarily a change of enrichment for enriched Mackey functors (presheaves). The essential reason is that symmetry of a multifunctor is required for commuting the opposite construction in the domain of enriched presheaves with change of enrichment. Sections 10.5 and 10.6 give applications to Elmendorf–Mandell K-theory, with attention to the relevant symmetry conditions among other details.