This chapter describes the theory of self-enrichment for closed multicategories, and of standard enrichment for multifunctors between closed multicategories. The self-enrichment of the multicategory of permutative categories, from Chapter 8, is a special case. Compositionality of standard enrichment is discussed in Section 9.3, and applied to the factorization of Elmendorf–Mandell K-theory in Section 9.4.

This chapter reviews the K-theory functors due to Segal and Elmendorf–Mandell. These are also called infinite loop space machines because they produce connective spectra from permutative categories and multicategories. Each is constructed as a composite of other functors, via certain diagram categories, that we describe.

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.