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2.1 The Theory of $2$-Categories

The collection of (small) categories can itself be organized into a (large) category $\operatorname{Cat}$, whose objects are small categories and whose morphisms are functors. However, the structure of $\operatorname{Cat}$ as an abstract category fails to capture many of the essential features of category theory:

$(i)$

Given a pair of functors $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ with the same source and target, we are usually not interested in the question of whether or not $F$ and $G$ are equal. Instead, we should regard $F$ and $G$ as interchangeable if there exists a natural isomorphism $\alpha : F \simeq G$. This sort of information is not encoded in the structure of the category $\operatorname{Cat}$.

$(ii)$

Given a pair of categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, we are usually not interested in the question of whether or not $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are isomorphic. Instead, we should regard $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ as interchangeable if there exists an equivalence of categories from $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. In this case, the functor $F$ need not be invertible when regarded as a morphism in $\operatorname{Cat}$.

To remedy the situation, it is useful to contemplate a more elaborate structure.

Definition 2.1.0.1. A strict $2$-category $\operatorname{\mathcal{C}}$ consists of the following data:

  • A collection of objects $\{ X, Y, Z, \cdots \} $; we will write $X \in \operatorname{\mathcal{C}}$ to indicate that $X$ is an object of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $X, Y \in \operatorname{\mathcal{C}}$, a category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y)$. We refer to objects $f$ of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ as $1$-morphisms from $X$ to $Y$ and write $f: X \rightarrow Y$ to indicate that $f$ is a $1$-morphism from $X$ to $Y$. Given a pair of $1$-morphisms $f,g \in \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$, we refer to morphisms from $f$ to $g$ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ as $2$-morphisms from $f$ to $g$.

  • For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, a composition functor

    \[ \circ : \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( Y, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z). \]
  • For every object $X \in \operatorname{\mathcal{C}}$, an identity $1$-morphism $\operatorname{id}_{X} \in \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$.

These data are required to satisfy the following conditions:

$(1)$

For each object $X \in \operatorname{\mathcal{C}}$, the identity $1$-morphism $\operatorname{id}_{X}$ is a unit for both right and left composition. That is, for every object $Y \in \operatorname{\mathcal{C}}$, the functors

\[ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \quad \quad f \mapsto f \circ \operatorname{id}_{X} \]
\[ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,X) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,X) \quad \quad g \mapsto \operatorname{id}_{X} \circ g \]

are both equal to the identity.

$(2)$

The composition law of $\operatorname{\mathcal{C}}$ is strictly associative. That is, for every quadruple of objects $W, X, Y, Z \in \operatorname{\mathcal{C}}$, the diagram of categories

\[ \xymatrix { \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( Y, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X,Y) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( W, X) \ar [r]^-{\operatorname{id}\times \circ } \ar [d]^{\circ \times \operatorname{id}} & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( Y, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( W, Y) \ar [d]^{ \circ } \\ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( W, X) \ar [r]^-{\circ } & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( W, Z) } \]

commutes (in the ordinary category $\operatorname{Cat}$).

Example 2.1.0.2. We define a strict $2$-category $\mathbf{Cat}$ as follows:

  • The objects of $\mathbf{Cat}$ are (small) categories.

  • For every pair of small categories $\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}\in \mathbf{\operatorname{Cat}}$, we take $\underline{\operatorname{Hom}}_{ \mathbf{Cat} }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ to be the category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ of functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.

  • The composition law on $\mathbf{Cat}$ is given by the usual composition of functors.

We will refer to $\mathbf{Cat}$ as the strict $2$-category of (small) categories.

The reader might at this point object that the definition of strict $2$-category violates a fundamental principle of category theory: axioms $(a)$ and $(b)$ of Definition 2.1.0.1 require that certain functors are equal. In practice, one often encounters mathematical structures $\operatorname{\mathcal{C}}$ which do not quite fit in the framework of Definition 2.1.0.1, because the associative law for composition of $1$-morphisms in $\operatorname{\mathcal{C}}$ holds only up to isomorphism. To address this point, Bénabou introduced a more general type of structure which he called a bicategory, which we will refer to here as a $2$-category.

Our goal in this section is to give a brief introduction to the theory of $2$-categories. We begin in §2.1.1 by reviewing the definition of a $2$-category (Definition 2.1.1.1) and establishing some notational and terminological conventions. Every strict $2$-category can be regarded as a $2$-category (Example 2.1.1.4), but many of the $2$-categories which arise “in nature” fail to be strict: we discuss several examples of this phenomenon in §2.1.2.

To articulate the relationship between $2$-categories and strict $2$-categories more precisely, it is convenient to view each as the objects of a suitable (ordinary) category. In §2.1.4, we introduce the notion of a functor between $2$-categories (Definition 2.1.4.3). Roughly speaking, a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an operation which carries objects, $1$-morphisms, and $2$-morphisms of $\operatorname{\mathcal{C}}$ to objects, $1$-morphisms, and $2$-morphisms of $\operatorname{\mathcal{D}}$, which is compatible with the composition laws on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$. Here again there are several possible definitions, depending on whether one demands that the compatibility holds strictly (in which case we say that $F$ is a strict functor), up to isomorphism (in which case we say that $F$ is a functor), or up to possible non-invertible $2$-morphism (in which case we say that $F$ is a lax functor). We use this notion in §2.1.5 to introduce an (ordinary) category $\operatorname{2Cat}$, whose objects are $2$-categories and whose morphisms are functors between $2$-categories (and consider several other variations on this theme).

The notion of $2$-category is more general than the notion of strict $2$-category defined above: in general, a $2$-category $\operatorname{\mathcal{C}}$ need not be strict or even isomorphic (as an object of $\operatorname{2Cat}$) to a strict $2$-category $\operatorname{\mathcal{C}}'$. However, we will prove in §2.1.6 that every $2$-category $\operatorname{\mathcal{C}}$ is isomorphic to a strictly unitary $2$-category $\operatorname{\mathcal{C}}'$ (Example 2.1.6.14): that is, a $2$-category $\operatorname{\mathcal{C}}'$ in which the composition law is strictly unital, but not necessarily strictly associative (see Variant 2.1.1.5). We will use this observation at various points to simplify our analysis of the Duskin nerve in §2.2.

Remark 2.1.0.3. Let $\operatorname{\mathcal{C}}$ be a $2$-category. It is generally not possible to find a strict $2$-category $\operatorname{\mathcal{C}}'$ which is isomorphic to $\operatorname{\mathcal{C}}$ (as an object of the category $\operatorname{2Cat}$ we will introduce in §2.1.5). However, it is always possibly to find a strict $2$-category $\operatorname{\mathcal{C}}'$ which is equivalent to $\operatorname{\mathcal{C}}$; we will return to this point in §.

Structure

  • Subsection 2.1.1: $2$-Categories
  • Subsection 2.1.2: Examples of $2$-Categories
  • Subsection 2.1.3: Opposite and Conjugate $2$-Categories
  • Subsection 2.1.4: Functors of $2$-Categories
  • Subsection 2.1.5: The Category of $2$-Categories
  • Subsection 2.1.6: Isomorphisms of $2$-Categories