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 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 applies the general theory from Chapter 11 to change of enrichment along the inverse equivalences of homotopy theories developed in Part 2. The main results, Theorems 12.1.6, 12.4.6, and 12.6.6, establish equivalences of homotopy theories for enriched diagram categories and Mackey functor categories over pointed multicategories, permutative categories, and M1-modules.

This chapter extends the general multicategorical enrichment theory from Chapter 6 to enrichment over closed multicategories. Because enrichment over permutative categories is both illustrative of the general theory and essential for the further applications, this chapter focuses on that case in detail.