2.2.5 The Category of Bicategories
We now show that bicategories can be regarded as the objects of a category $\operatorname{2Cat}$, in which the morphisms are functors between bicategories (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 bicategories, 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 bicategory $\operatorname{\mathcal{C}}$, 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 bicategories, 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 bicategory. 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) bicategories and whose morphisms are lax functors between bicategories (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 bicategories, and the morphisms of $\operatorname{2Cat}$ are functors.
The objects of $\operatorname{2Cat}_{\operatorname{Str}}$ are strict bicategories, and the morphisms of $\operatorname{2Cat}_{\operatorname{Str}}$ are strict functors.
The objects of $\operatorname{2Cat}_{\operatorname{ULax}}$ are bicategories, and the morphisms of $\operatorname{2Cat}_{\operatorname{ULax}}$ are strictly unital lax functors.
We will refer to $\operatorname{2Cat}$ as the category of bicategories, and to $\operatorname{2Cat}_{\operatorname{Str}}$ as the category of strict bicategories.
Example 2.2.5.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be ordinary categories, which we regard as bicategories having only identity $2$-morphisms (see Example 2.2.0.6). Then every lax functor of bicategories 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 bicategories. 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.
