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2.1.5 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 2.1.5.5). There are several variants of this construction, depending on what sort of functors we allow.

Construction 2.1.5.1 (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 2.1.5.2. 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 2.1.5.1 are compatible with the unit constraints and associativity constraints of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}$, as required by Definition 2.1.4.3.

Remark 2.1.5.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be lax functors of $2$-categories, and let $GF: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ be their composition. Then:

  • If $F$ and $G$ are unitary, then the composition $GF$ is unitary.

  • If $F$ and $G$ are functors, then the composition $GF$ is a functor.

  • If $F$ and $G$ are strictly unitary, then the composition $GF$ is strictly unitary.

  • If $F$ and $G$ are strict functors, then the composition $GF$ is a strict functor.

Example 2.1.5.4. 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 2.1.5.1.

Definition 2.1.5.5. 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 2.1.4.3), with composition given by Construction 2.1.5.1 and identity morphisms given by Example 2.1.5.4. 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.

Variant 2.1.5.6. Let $\operatorname{2Cat}'_{\operatorname{ULax}}$ denote the subcategory of $\operatorname{2Cat}_{\operatorname{Lax}}$ (and full subcategory of $\operatorname{2Cat}_{\operatorname{ULax}}$) whose objects are strictly unitary $2$-categories and whose morphisms are strictly unitary lax functors. We will see in §2.1.6 that the inclusion $\operatorname{2Cat}'_{\operatorname{ULax}} \hookrightarrow \operatorname{2Cat}_{\operatorname{ULax}}$ is an equivalence of categories: that is, every $2$-category $\operatorname{\mathcal{C}}$ is isomorphic (via a strictly unitary lax functor) to a strictly unitary $2$-category (Example 2.1.6.14).

Remark 2.1.5.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories. Then the collection $\operatorname{Hom}_{ \operatorname{2Cat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ of functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ can be identified with the set of objects of a certain $2$-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, called the $2$-category of functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$. We will return to this point in more detail in §.

Example 2.1.5.8. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be ordinary categories, which we regard as $2$-categories having only identity $2$-morphisms (see Example 2.1.1.7). Then every lax functor of $2$-categories from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is automatically strict (Example 2.1.4.7), 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 2.1.1.7 as supplying fully faithful embeddings (of ordinary categories)

\[ \operatorname{Cat}\hookrightarrow \operatorname{2Cat}_{\operatorname{Str}} \quad \quad \operatorname{Cat}\hookrightarrow \operatorname{2Cat}\quad \quad \operatorname{Cat}\hookrightarrow \operatorname{2Cat}_{\operatorname{Lax}} \quad \quad \operatorname{Cat}\hookrightarrow \operatorname{2Cat}_{\operatorname{ULax}}. \]

Remark 2.1.5.9 (Functors on Opposite $2$-Categories). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories, and let $\operatorname{\mathcal{C}}^{\operatorname{op}}$ and $\operatorname{\mathcal{D}}^{\operatorname{op}}$ denote their opposites (Construction 2.1.3.1). Then every lax functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ induces a lax functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$, given explicitly by the formulae

\[ F^{\operatorname{op}} (X^{\operatorname{op}} ) = F(X)^{\operatorname{op}} \quad \quad F^{\operatorname{op}}(f^{\operatorname{op}} ) = F(f)^{\operatorname{op}} \quad \quad F^{\operatorname{op}}( \gamma ^{\operatorname{op}} ) = F(\gamma )^{\operatorname{op}} \]
\[ \epsilon _{X^{\operatorname{op}}} = (\epsilon _{X})^{\operatorname{op}} \quad \quad \mu _{ g^{\operatorname{op}}, f^{\operatorname{op}} } = (\mu _{f,g})^{\operatorname{op}}. \]

In this case, $F$ is a functor if and only if $F^{\operatorname{op}}$ is a functor, and a strict functor if and only if $F^{\operatorname{op}}$ is a strict functor. This operation is compatible with composition, and therefore induces equivalences of categories

\[ \operatorname{2Cat}_{\operatorname{Str}} \simeq \operatorname{2Cat}_{\operatorname{Str}} \quad \quad \operatorname{2Cat}\simeq \operatorname{2Cat}\quad \quad \operatorname{2Cat}_{\operatorname{Lax}} \simeq \operatorname{2Cat}_{\operatorname{Lax}} \quad \quad \operatorname{2Cat}_{\operatorname{ULax}} \simeq \operatorname{2Cat}_{\operatorname{ULax}}. \]

Remark 2.1.5.10 (Functors on Conjugate $2$-Categories). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories, and let $\operatorname{\mathcal{C}}^{\operatorname{c}}$ and $\operatorname{\mathcal{D}}^{\operatorname{c}}$ denote their conjugates (Construction 2.1.3.3). Then every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ induces a functor $F^{\operatorname{c}}: \operatorname{\mathcal{C}}^{\operatorname{c}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{c}}$, given explicitly by the formulae

\[ F^{\operatorname{c}} (X^{\operatorname{c}} ) = F(X)^{\operatorname{c}} \quad \quad F^{\operatorname{c}}(f^{\operatorname{c}} ) = F(f)^{\operatorname{c}} \quad \quad F^{\operatorname{c}}( \gamma ^{\operatorname{c}} ) = F(\gamma )^{\operatorname{c}} \]
\[ \epsilon _{X^{\operatorname{c}}} = (\epsilon ^{-1}_{X})^{\operatorname{c}} \quad \quad \mu _{ g^{\operatorname{c}}, f^{\operatorname{c}} } = (\mu ^{-1}_{g,f})^{\operatorname{c}}. \]

In this case, the functor $F$ is strict if and only if $F^{\operatorname{c}}$ is strict. This operation is compatible with composition, and therefore induces equivalences of categories

\[ \operatorname{2Cat}_{\operatorname{Str}} \simeq \operatorname{2Cat}_{\operatorname{Str}} \quad \quad \operatorname{2Cat}\simeq \operatorname{2Cat} \]

Warning 2.1.5.11. The construction of Remark 2.1.5.10 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.

Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a (lax) functor between $2$-categories. According to Example 2.1.4.5, $F$ is strict if and only if the identity and composition constraints

\[ \epsilon _{X}: \operatorname{id}_{F(X)} \Rightarrow F( \operatorname{id}_ X ) \quad \quad \mu _{g,f}: F(g) \circ F(f) \Rightarrow F( g \circ f) \]

are identity $2$-morphisms in $\operatorname{\mathcal{D}}$. In §2.2.1, it will be useful to consider a weaker version of this condition, where we require strict compatibility with the formation of identity morphisms but not with respect to composition in general.