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 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 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 shows that the pointed free construction from Chapter 4 is a nonsymmetric multifunctor. Furthermore, it provides equivalences of homotopy theories between categories of nonsymmetric algebras in pointed multicategories and permutative categories. This is the basis for applications to enriched diagrams in Chapter 12.