The Yoneda Lemma and Coherence

In this chapter, the Yoneda Lemma and the Coherence Theorem for bicategories are stated and proved. The chapter discusses the bicategorical Yoneda pseudofunctor, a bicategorical version of the Yoneda embedding for a bicategory, which is a local equivalence, and the Bicategorical Yoneda Lemma. A consequence of the Bicategorical Whitehead Theorem and the Bicategorical Yoneda Embedding is the Bicategorical Coherence Theorem, which states that every bicategory is biequivalent to a 2-category.

Functors, Transformations, and Modifications

This chapter discusses functors, transformations, and modifications that are bicategorical analogs of functors and natural transformations. The main concepts covered are lax functors, lax transformations, modifications, and icons. A section is devoted to representable pseudofunctors, representable transformations, and representable modifications, which will be used in the Bicategorical Yoneda Lemma.

The Tricategory of Bicategories

In this chapter, the tricategory of bicategories is presented in full detail. After a preliminary discussion of the whiskerings of a lax transformation with a lax functor, the chapter goes on to define a tricategory. The rest of the chapter proves in detail the existence of a tricategory with small bicategories as objects (i.e. a tricategory of bicategories), pseudofunctors as 1-cells, strong transformations as 2-cells, and modifications as 3-cells.

Pasting Diagrams

In this chapter, pasting diagrams are defined, and pasting theorems for 2-/bicategories are proved. Pasting is an essential reasoning tool in 2-dimensional category theory. Each pasting theorem says that a pasting diagram, in a 2-category or a bicategory, has a unique composite. String diagrams, which provide another way to visualize and manipulate pasting diagrams, are also discussed.

The Grothendieck Construction

This chapter defines the Grothendieck construction for a lax functor into the category of small categories. It then proves that, for such a pseudofunctor, its Grothendieck construction is its lax colimit. Most of the rest of the chapter contains a detailed proof of the Grothendieck Construction Theorem, which states that the Grothendieck construction is part of a 2-equivalence. A generalization of the Grothendieck construction that applies to an indexed bicategory is also discussed.

Further 2-Dimensional Categorical Structures

In this chapter, further 2-dimensional categorical structures are presented and discussed. These include monoidal bicategories, as one-object tricategories, along with braided monoidal bicategories, sylleptic monoidal bicategories, and symmetric monoidal bicategories. The rest of the chapter discusses the Gray tensor product on 2-categories, Gray monoids, double categories, and monoidal double categories.

Grothendieck Fibrations

In this chapter, Grothendieck fibrations are defined, and the Grothendieck Fibration Theorem is proved. After discussing some basic definitions, properties, and examples of fibrations, this chapter constructs a 2-monad and proves in detail that its pseudo algebras are precisely cloven fibrations. Moreover, the strict algebras of this 2-monad are shown to be precisely split fibrations.

Bicategories and 2-Categories

In this chapter, 2-categories and bicategories are defined, along with basic examples. Several useful unity properties in bicategories, generalizing those in monoidal categories and underlying many fundamental results in bicategory theory, are discussed. In addition to well-known examples, the 2-categories of multicategories and of polycategories are constructed. This chapter ends with a discussion of duality of bicategories.

Bicategorical Limits and Nerves

This chapter focuses on bicategorical limits and nerves, which are the 2-/bicategorical analogues of (co)limits and nerves. The chapter proves that lax limits, lax bilimits, pseudo limits, and pseudo bilimits are unique up to an equivalence and an invertible modification. The chapter discusses 2-limits and 2-colimits, as well as the Duskin nerve and the 2-nerve, which are two different generalizations of the 1-categorical Grothendieck nerve. In each case, an explicit description of the simplices is provided.

The Whitehead Theorem for Bicategories

In this chapter, the Whitehead Theorem for bicategories is proved in detail. The Whitehead Theorem states that a pseudofunctor between bicategories is a biequivalence if and only if it is surjective up to adjoint equivalences on objects, surjective up to isomorphisms on 1-cells, and bijective on 2-cells. The chapter covers the lax slice bicategory, lax terminal objects, and the Quillen Theorem A for bicategories. A 2-categorical version of the Whitehead Theorem is also discussed.