2.2 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 mathematical structure.
Definition 2.2.0.1. A strict $2$-category $\operatorname{\mathcal{C}}$ consists of the following data:
A collection $\operatorname{Ob}(\operatorname{\mathcal{C}})$, whose elements we refer to as objects of $\operatorname{\mathcal{C}}$. We will often abuse notation by writing $X \in \operatorname{\mathcal{C}}$ to indicate that $X$ is an element of $\operatorname{Ob}(\operatorname{\mathcal{C}})$.
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
\[ \xymatrix@R =50pt@C=50pt{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( Y, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X,Y) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( W, X) \ar [r]^-{\operatorname{id}\times \circ } \ar [d]^{\circ \times \operatorname{id}} & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( Y, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( W, Y) \ar [d]^{ \circ } \\ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( W, X) \ar [r]^-{\circ } & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( W, Z) } \]
commutes (in the ordinary category $\operatorname{Cat}$).
Example 2.2.0.4. We define a strict $2$-category $\mathbf{Cat}$ as follows:
The objects of $\mathbf{Cat}$ are (small) categories.
For every pair of small categories $\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}\in \mathbf{Cat}$, 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. Note that the underlying ordinary category of $\mathbf{Cat}$ is the category $\operatorname{Cat}$ (whose objects are small categories and morphisms are functors).
We can obtain many more examples by studying categories equipped with additional structure.
Example 2.2.0.5. We define a strict $2$-category $\mathbf{MonCat}$ as follows:
The objects of $\mathbf{MonCat}$ are (small) monoidal categories.
For every pair of small monoidal categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, we take $\underline{\operatorname{Hom}}_{ \mathbf{MonCat} }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ to be the category $\operatorname{Fun}^{\otimes }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ of monoidal functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ (Notation 2.1.6.9).
The composition law on $\mathbf{MonCat}$ is given by the composition of monoidal functors described in Remark 2.1.6.13.
There are several obvious variants on this construction: for example, we can work with nonunital monoidal categories in place of monoidal categories, or lax monoidal functors in place of monoidal functors.
Example 2.2.0.6 (Ordinary Categories). Every ordinary category can be regarded as a strict $2$-category. More precisely, to each category $\operatorname{\mathcal{C}}$ we can associate a strict $2$-category $\operatorname{\mathcal{C}}'$ as follows:
The objects of $\operatorname{\mathcal{C}}'$ are the objects of $\operatorname{\mathcal{C}}$.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, objects of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X,Y)$ are elements of the set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$, and every morphism in $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X,Y)$ is an identity morphism.
For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the composition functor
\[ \circ : \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}( Y, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X,Z) \]
is given on objects by the composition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$.
For every object $X \in \operatorname{\mathcal{C}}$, the identity object $\operatorname{id}_{X} \in \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X,X)$ coincides with the identity morphism $\operatorname{id}_ X \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)$.
In this situation, we will generally abuse terminology by identifying the strict $2$-category $\operatorname{\mathcal{C}}'$ with the ordinary category $\operatorname{\mathcal{C}}$ (see Example 2.2.5.7).
Example 2.2.0.8 (Delooping). Let $\operatorname{\mathcal{M}}$ be a category equipped with a strict monoidal structure $\otimes : \operatorname{\mathcal{M}}\times \operatorname{\mathcal{M}}\rightarrow \operatorname{\mathcal{M}}$ (Definition 2.1.2.1). We define a strict $2$-category $B\operatorname{\mathcal{M}}$ as follows:
The set of objects $\operatorname{Ob}( B\operatorname{\mathcal{M}})$ is the singleton set $\{ X \} $.
The category $\underline{\operatorname{Hom}}_{B\operatorname{\mathcal{M}}}( X, X )$ is equal to $\operatorname{\mathcal{M}}$.
The composition functor $\circ : \underline{\operatorname{Hom}}_{B\operatorname{\mathcal{M}}}( X, X ) \times \underline{\operatorname{Hom}}_{B\operatorname{\mathcal{M}}}( X, X ) \rightarrow \underline{\operatorname{Hom}}_{B\operatorname{\mathcal{M}}}( X, X )$ is equal to the tensor product $\otimes : \operatorname{\mathcal{M}}\times \operatorname{\mathcal{M}}\rightarrow \operatorname{\mathcal{M}}$.
The identity morphism $\operatorname{id}_{X}$ is the strict unit object of $\operatorname{\mathcal{M}}$.
We will refer to $B\operatorname{\mathcal{M}}$ as the delooping of $\operatorname{\mathcal{M}}$.
Note that the constructions
\[ \operatorname{\mathcal{M}}\mapsto B\operatorname{\mathcal{M}}\quad \quad \operatorname{\mathcal{C}}\mapsto \underline{\operatorname{End}}_{\operatorname{\mathcal{C}}}(X) \]
induce mutually inverse bijections
\[ \{ \text{Strict Monoidal Categories $\operatorname{\mathcal{M}}$} \} \simeq \{ \text{Strict $2$-Categories $\operatorname{\mathcal{C}}$ with $\operatorname{Ob}(\operatorname{\mathcal{C}}) = \{ X \} $} \} , \]
generalizing the identification of Remark 1.3.2.4.
The reader might at this point object that the definition of strict $2$-category violates a fundamental principle of category theory: axioms $(1)$ and $(2)$ of Definition 2.2.0.1 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 2.2.0.1, 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 §2.2.1 by reviewing the definition of a $2$-category (Definition 2.2.1.1) and establishing some notational and terminological conventions. Every strict $2$-category can be regarded as a $2$-category (Example 2.2.1.4), but many of the $2$-categories which arise “in nature” fail to be strict: we discuss several examples of this phenomenon in §2.2.2.
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 §2.2.4, we introduce the notion of a functor between $2$-categories (Definition 2.2.4.5). 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 §2.2.5 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 §2.2.7 that every $2$-category $\operatorname{\mathcal{C}}$ is isomorphic to a strictly unitary $2$-category $\operatorname{\mathcal{C}}'$: that is, a $2$-category $\operatorname{\mathcal{C}}'$ in which the composition law is strictly unital, but not necessarily strictly associative (Proposition 2.2.7.7). The proof will make use of a certain twisting procedure in the setting of $2$-categories (Construction 2.2.6.8), which we will describe in 2.2.6.
Structure
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Subsection 2.2.1: $2$-Categories
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Subsection 2.2.2: Examples of $2$-Categories
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Subsection 2.2.3: Opposite and Conjugate $2$-Categories
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Subsection 2.2.4: Functors of $2$-Categories
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Subsection 2.2.5: The Category of $2$-Categories
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Subsection 2.2.6: Isomorphisms of $2$-Categories
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Subsection 2.2.7: Strictly Unitary $2$-Categories
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Subsection 2.2.8: The Homotopy Category of a $2$-Category
