This chapter reviews the 2-category of small multicategories, including three important special cases. These are pointed multicategories, left M1-modules, and permutative categories with multilinear functors. These variants are related by various free, forgetful, and endomorphism functors that will be used throughout the rest of this work.

This chapter reviews equivalences of homotopy theories between Multicat, the category of small multicategories and multifunctors, PermCat^st, the category of small permutative categories and strict monoidal functors, and PermCat^su, the category of small permutative categories and strictly unital symmetric monoidal functors. These equivalences are given by a free left adjoint to the endomorphism functor. This material provides an important foundation for that of Part 2.

This chapter extends the material of Chapter 3 to a pointed free construction from pointed multicategories to permutative categories. This is not a restriction, along the inclusion of pointed multicategories among all multicategories, but an extension, along the functor that adjoins a disjoint basepoint. Essential results, such as the adjunction with the endomorphism construction and compatibility with stable equivalences, are likewise extended from Chapter 3.

This chapter establishes the general theory for a pair of nonsymmetric multifunctors (E,F) to provide inverse equivalences of homotopy theories between enriched diagram categories. The main result is Theorem 11.4.14 and does not require E or F to satisfy the symmetry condition of a multifunctor. A similar result for enriched Mackey functor categories, in Theorem 11.4.24, requires that E, but not necessarily F, is a multifunctor. This is important for the applications, Theorems 12.1.6 and 12.4.6. There, E is an endomorphism multifunctor and F is a corresponding free nonsymmetric multifunctor.

This chapter gives the basic definitions and results for enrichment in a nonsymmetric multicategory M. Proposition 6.2.1 shows that this material agrees, in the case that M is the endomorphism multicategory of a monoidal category, with classical enriched category theory over V. The main application takes M to be the multicategory of permutative categories and strictly unital symmetric monoidal functors, which is not an endomorphism multicategory. Sections 6.3 through 6.5 treat this case in detail.

This chapter develops the first collection of results around change of enrichment along a (possibly nonsymmetric) multifunctor. As in Chapter 6, it is shown that this theory extends the classical theory for enrichment over (possibly symmetric) monoidal categories. Compositionality and 2-functoriality for the change-of-enrichment constructions are treated in Sections 7.4 and 7.5, respectively.

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.