2.2.5 The Category of $2$-Categories
We now show that $2$-categories can be regarded as the objects of a category $\operatorname{2Cat}$, in which the morphisms are functors between $2$-categories (Definition 2.2.5.5). There are several variants of this construction, depending on what sort of functors we allow.
Construction 2.2.5.1 (Composition of Lax Functors). Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be $2$-categories, and suppose we are given a pair of lax functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$. We define a lax functor $GF: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ as follows:
On objects, the lax functor $GF$ is given by $(GF)(X) = G( F(X) )$.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the functor
\[ (GF)_{X,Y}: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y ) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{E}}}( (GF)(X), (GF)(Y) ) \]
is given by the composition of functors
\[ \underline{ \operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow { F_{X,Y} } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \xrightarrow { G_{F(X), F(Y)} } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{E}}}( (GF)(X), (GF)(Y) ). \]
In other words, the lax functor $GF$ is given on $1$-morphisms and $2$-morphisms by the formulae
\[ (GF)(f) = G( F(f) ) \quad \quad (GF)( \gamma ) = G( F( \gamma ) ). \]
For each object $X \in \operatorname{\mathcal{C}}$, the identity constraint $\epsilon _{X}^{GF}: \operatorname{id}_{ (GF)(X)} \Rightarrow (GF)(\operatorname{id}_ X)$ is given by the composition
\[ \operatorname{id}_{(GF)(X)} \xRightarrow { \epsilon ^{G}_{F(X)} } G( \operatorname{id}_{F(X)} ) \xRightarrow { G( \epsilon ^{F}_{X} ) } (GF)(\operatorname{id}_ X). \]
For every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ in the $2$-category $\operatorname{\mathcal{C}}$, the composition constraint $\mu _{g,f}^{GF}: (GF)(g) \circ (GF)(f) \rightarrow (GF)( g \circ f )$ is given by the composition
\[ (GF)(g) \circ (GF)(f) \xRightarrow { \mu ^{G}_{F(g), F(f)} } G( F(g) \circ F(f) ) \xRightarrow { G( \mu ^{F}_{g,f} ) } (GF)( g \circ f). \]
We will refer to $GF$ as the composition of $F$ with $G$, and will sometimes denote it by $G \circ F$.
Exercise 2.2.5.2. Check that the composition of lax functors is well-defined. That is, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ are lax functors between $2$-categories, then the identity and composition constraints $\epsilon ^{GF}_{X}$ and $\mu ^{GF}_{g,f}$ of Construction 2.2.5.1 are compatible with the unit constraints and associativity constraints of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}$, as required by Definition 2.2.4.5.
Example 2.2.5.4. Let $\operatorname{\mathcal{C}}$ be a $2$-category. We let $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ be the strict functor which carries every object, $1$-morphism, and $2$-morphism of $\operatorname{\mathcal{C}}$ to itself. We will refer to $\operatorname{id}_{\operatorname{\mathcal{C}}}$ as the identity functor on $\operatorname{\mathcal{C}}$. Note that it is both a left and right unit for the composition of lax functors given in Construction 2.2.5.1.
Definition 2.2.5.5. We let $\operatorname{2Cat}_{\operatorname{Lax}}$ denote the ordinary category whose objects are (small) $2$-categories and whose morphisms are lax functors between $2$-categories (Definition 2.2.4.5), with composition given by Construction 2.2.5.1 and identity morphisms given by Example 2.2.5.4. We define (non-full) subcategories
\[ \operatorname{2Cat}_{\operatorname{Str}} \subsetneq \operatorname{2Cat}\subsetneq \operatorname{2Cat}_{\operatorname{Lax}} \supsetneq \operatorname{2Cat}_{\operatorname{ULax}} \]
The objects of $\operatorname{2Cat}$ are $2$-categories, and the morphisms of $\operatorname{2Cat}$ are functors.
The objects of $\operatorname{2Cat}_{\operatorname{Str}}$ are strict $2$-categories, and the morphisms of $\operatorname{2Cat}_{\operatorname{Str}}$ are strict functors.
The objects of $\operatorname{2Cat}_{\operatorname{ULax}}$ are $2$-categories, and the morphisms of $\operatorname{2Cat}_{\operatorname{ULax}}$ are strictly unitary lax functors.
We will refer to $\operatorname{2Cat}$ as the category of $2$-categories, and to $\operatorname{2Cat}_{\operatorname{Str}}$ as the category of strict $2$-categories.
Example 2.2.5.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be ordinary categories, which we regard as $2$-categories having only identity $2$-morphisms (see Example 2.2.0.6). Then every lax functor of $2$-categories from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is automatically strict (Example 2.2.4.14), and can be identified with a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ in the usual sense. In other words, we can view Example 2.2.0.6 as supplying fully faithful embeddings (of ordinary categories)
\[ \operatorname{Cat}\hookrightarrow \operatorname{2Cat}_{\operatorname{Str}} \quad \quad \operatorname{Cat}\hookrightarrow \operatorname{2Cat}\quad \quad \operatorname{Cat}\hookrightarrow \operatorname{2Cat}_{\operatorname{Lax}} \quad \quad \operatorname{Cat}\hookrightarrow \operatorname{2Cat}_{\operatorname{ULax}}. \]
Warning 2.2.5.11. The construction of Remark 2.2.5.10 requires that the identity and composition constraints of $F$ are invertible, and therefore does not extend to lax functors between $2$-categories. In general, one cannot identify lax functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ with lax functors from $\operatorname{\mathcal{C}}^{\operatorname{c}}$ to $\operatorname{\mathcal{D}}^{\operatorname{c}}$: the definition of lax functor is asymmetrical with respect to vertical composition.
