2.2.4 Functors of $2$-Categories
Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories. Roughly speaking, a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ should be 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 suitably compatible with (horizontal and vertical) composition. Here it is useful to distinguish between different notions of functor, which are differentiated by the degree of compatibility which is assumed.
Definition 2.2.4.1 (Strict Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories. A strict functor $F$ from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ consists of the following data:
For every object $X \in \operatorname{\mathcal{C}}$, an object $F(X)$ in $\operatorname{\mathcal{D}}$.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, a functor of ordinary categories
\[ F_{X,Y}: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}(F(X),F(Y)). \]
We will generally abuse notation by writing $F(f)$ for the value of the functor $F_{X,Y}$ on an object $f$ of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$, and $F(\gamma )$ for the value of $F$ on a morphism $\gamma $ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$.
This data is required to satisfy the following compatibility conditions:
- $(1)$
For every object $X \in \operatorname{\mathcal{C}}$, we have $\operatorname{id}_{F(X)} = F( \operatorname{id}_ X )$.
- $(2)$
For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the diagram of categories
\[ \xymatrix@R =50pt@C=50pt{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \ar [r]^-{\circ } \ar [d]^{F_{Y,Z} \times F_{X,Y}} & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d]^{F_{X,Z}} \\ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}(F(Y),F(Z)) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}(F(X),F(Y)) \ar [r]^-{ \circ } & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Z) ) } \]
is strictly commutative.
- $(3)$
For every object $X \in \operatorname{\mathcal{C}}$, the functor $F_{X,X}$ carries the unit constraint $\upsilon _{X}: \operatorname{id}_{X} \circ \operatorname{id}_{X} \xRightarrow {\sim } \operatorname{id}_{X}$ to the unit constraint $\upsilon _{F(X)}: \operatorname{id}_{F(X)} \circ \operatorname{id}_{F(X)} \xRightarrow {\sim } \operatorname{id}_{F(X)}$.
- $(4)$
For every composable triple of $1$-morphisms $W \xrightarrow {f} X \xrightarrow {g} Y \xrightarrow {h} Z$ in $\operatorname{\mathcal{C}}$, we have $F( \alpha _{h,g,f} ) = \alpha _{F(h), F(g), F(f) }$. In other words, $F$ carries the associativity constraints of $\operatorname{\mathcal{C}}$ to the associativity constraints of $\operatorname{\mathcal{D}}$.
Example 2.2.4.3. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be strict $2$-categories, which we regard as $\operatorname{Cat}$-enriched categories (Remark 2.2.0.2). Then strict functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ (in the sense of Definition 2.2.4.1) can be identified with $\operatorname{Cat}$-enriched functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ (in the sense of Definition 2.1.7.10).
Exercise 2.2.4.4. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a strict functor. Show that, for each morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, the functor $F_{X,Y}: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ carries the left and right unit constraints $\lambda _{f}: \operatorname{id}_{Y} \circ f \xRightarrow {\sim } f$ and $\rho _{f}: f \circ \operatorname{id}_{X} \xRightarrow {\sim } f$ to $\lambda _{F(f) }$ and $\rho _{F(f)}$, respectively (see Construction 2.2.1.11).
Note that axiom $(2)$ of Definition 2.2.4.1 implies in particular that for every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ in the $2$-category $\operatorname{\mathcal{C}}$, we have an equality $F(g) \circ F(f) = F( g \circ f)$ between objects of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Z) )$. In practice, this requirement is often too strong: it is often better to allow a more liberal notion of functor, which is only required to preserve composition up to isomorphism.
Definition 2.2.4.5 (Lax Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories. A lax functor $F$ from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ consists of the following data:
For every object $X \in \operatorname{\mathcal{C}}$, an object $F(X) \in \operatorname{\mathcal{D}}$.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, a functor of ordinary categories
\[ F_{X,Y}: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}(F(X),F(Y)). \]
We will generally abuse notation by writing $F(f)$ for the value of the functor $F_{X,Y}$ on an object $f$ of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$, an $F(\gamma )$ for the value of $F$ on a morphism $\gamma $ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$.
For every object $X \in \operatorname{\mathcal{C}}$, a $2$-morphism $\epsilon _{X}: \operatorname{id}_{ F(X)} \Rightarrow F( \operatorname{id}_ X )$ in the $2$-category $\operatorname{\mathcal{D}}$, which we will refer to as the identity constraint.
For every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ in the $2$-category $\operatorname{\mathcal{C}}$, a $2$-morphism
\[ \mu _{g,f}: F(g) \circ F(f) \Rightarrow F( g \circ f ), \]
which we will refer to as the composition constraint. We require that, if the objects $X$, $Y$, and $Z$ are fixed, then the construction $(g,f) \mapsto \mu _{g,f}$ is functorial: that is, we can regard $\mu $ as a natural transformation of functors as indicated in the diagram
\[ \xymatrix@C =40pt@R=40pt{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \ar [r]^-{\circ } \ar [d]_{F_{Y,Z} \times F_{X,Y}} & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d]^{F_{X,Z}} \\ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}(F(Y),F(Z)) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}(F(X),F(Y)) \ar [r]^-{ \circ } \ar@ {=>}[]+<30pt,15pt>;+<150pt,45pt>_-{\mu } & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Z) ) } \]
This data is required to be compatible with the unit and associativity constraints of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ in the following sense:
- $(a)$
For every $1$-morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, the left unit constraint $\lambda _{F(f)}$ in $\operatorname{\mathcal{D}}$ is given by the vertical composition
\[ \operatorname{id}_{ F(Y) } \circ F(f) \xRightarrow { \epsilon _{Y} \circ \operatorname{id}_{F(f)} } F( \operatorname{id}_{Y} ) \circ F(f) \xRightarrow { \mu _{ \operatorname{id}_ Y, f} } F( \operatorname{id}_ Y \circ f ) \xRightarrow { F( \lambda _ f) } F(f). \]
- $(b)$
For every $1$-morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, the right unit constraint $\rho _{F(f)}$ in $\operatorname{\mathcal{D}}$ is given by the vertical composition
\[ F(f) \circ \operatorname{id}_{F(X)} \xRightarrow { \operatorname{id}_{F(f)} \circ \epsilon _{X} } F(f) \circ F( \operatorname{id}_ X ) \xRightarrow { \mu _{ f, \operatorname{id}_ X} } F( f \circ \operatorname{id}_ X ) \xRightarrow { F( \rho _ f) } F(f). \]
- $(c)$
For every triple of composable $1$-morphisms $W \xrightarrow {f} X \xrightarrow {g} Y \xrightarrow {h} Z$ in the $2$-category $\operatorname{\mathcal{C}}$, we have a commutative diagram
\[ \xymatrix@C =60pt@R=40pt{ F(h) \circ (F(g) \circ F(f) ) \ar@ {=>}[d]^{ \operatorname{id}_{F(h)} \circ \mu _{g,f} } \ar@ {=>}[r]^-{ \alpha _{F(h), F(g), F(f)} } & (F(h) \circ F(g) ) \circ F(f) \ar@ {=>}[d]^{ \mu _{h,g} \circ \operatorname{id}_{F(f)} } \\ F(h) \circ F( g \circ f) \ar@ {=>}[d]^{\mu _{h, g\circ f}} & F(h \circ g) \circ F(f) \ar@ {=>}[d]^{ \mu _{h \circ g, f} } \\ F( h \circ (g \circ f)) \ar@ {=>}[r]^-{ F( \alpha _{h,g,f})} & F( (h \circ g) \circ f) } \]
in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(W), F(Z) )$.
A functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a lax functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ with the property that 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 isomorphisms.
Warning 2.2.4.6. The terminology of Definition 2.2.4.5 is not standard. In [MR0220789], Bénabou uses the term morphism for what we call a lax functor of $2$-categories, homomorphism for what we call a functor of $2$-categories, and strict homomorphism for what we call a strict functor of $2$-categories. Other authors refer to functors of $2$-categories (in the sense of Definition 2.2.4.5) as weak functors or pseudofunctors (to avoid confusion with the notion of strict functor).
Example 2.2.4.10 (Algebras as Lax Functors). Let $[0]$ denote the category having a single object and a single morphism, which we regard as a (strict) $2$-category, and let $\operatorname{\mathcal{D}}$ be any $2$-category. Combining Remark 2.2.4.9 and Example 2.1.5.21, we deduce that lax functors $[0] \rightarrow \operatorname{\mathcal{D}}$ can be identified with pairs $(Y, A)$, where $Y \in \operatorname{\mathcal{D}}$ is an object and $A$ is an algebra object of the monoidal category $\underline{\operatorname{End}}_{\operatorname{\mathcal{D}}}(Y)$.
Example 2.2.4.11. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a strict functor (in the sense of Definition 2.2.4.1). Then we can regard $F$ as a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ (in the sense of Definition 2.2.4.5) by taking the identity and composition constraints
\[ \epsilon _ X: \operatorname{id}_{F(X)} \Rightarrow F( \operatorname{id}_ X) \quad \quad \mu _{g,f}: F(g) \circ F(f) \Rightarrow F( g \circ f) \]
to be the identity maps (note that in this case, conditions $(a)$, $(b)$, and $(c)$ of Definition 2.2.4.5 reduce to conditions $(3)$ and $(4)$ of Definition 2.2.4.1). Conversely, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a lax functor having the property that each of the identity and composition constraints $\epsilon _{X}$ and $\mu _{g,f}$ is an identity $2$-morphism of $\operatorname{\mathcal{D}}$, then we can regard $F$ as a strict $2$-functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$. We therefore have inclusions
\[ \{ \text{Strict functors $F:\operatorname{\mathcal{C}}\to \operatorname{\mathcal{D}}$} \} \subseteq \{ \text{Functors $F: \operatorname{\mathcal{C}}\to \operatorname{\mathcal{D}}$} \} \subseteq \{ \text{Lax functors $F: \operatorname{\mathcal{C}}\to \operatorname{\mathcal{D}}$} \} . \]
In general, neither of these inclusions is reversible.
Example 2.2.4.12 (Enriched Categories as Lax Functors). Let $S$ be a set, and let $\operatorname{\mathcal{E}}_{S}$ denote the indiscrete category with object set $S$: that is, the objects of $\operatorname{\mathcal{E}}_{S}$ are the elements of $S$, and $\operatorname{Hom}_{\operatorname{\mathcal{E}}_{S}}(X,Y)$ is a singleton for every pair of elements $X,Y \in S$. Regard $\operatorname{\mathcal{E}}_{S}$ as a (strict) $2$-category having only identity $2$-morphisms (Example 2.2.0.6). Let $\operatorname{\mathcal{C}}$ be a monoidal category, and let $B\operatorname{\mathcal{C}}$ be its delooping (Example 2.2.2.5). Unwinding the definitions, we see that lax functors $F: \operatorname{\mathcal{E}}_{S} \rightarrow B \operatorname{\mathcal{C}}$ (in the sense of Definition 2.2.4.5) can be identified with $\operatorname{\mathcal{C}}$-enriched categories having object set $S$ (in the sense of Definition 2.1.7.1).
Warning 2.2.4.13. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be strict $2$-categories, and let $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{D}}_0$ denote their underlying ordinary categories (obtained by ignoring the $2$-morphisms of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively). Every strict functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ induces a functor of ordinary categories $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}_0$. However, if a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is not strict, then it need not give rise to a functor from $\operatorname{\mathcal{C}}_0$ to $\operatorname{\mathcal{D}}_0$. If $X \xrightarrow {f} Y \xrightarrow {g} Z$ is a composable pair of $1$-morphisms in $\operatorname{\mathcal{C}}$, then Definition 2.2.4.5 guarantees that the $1$-morphisms $F(g) \circ F(f)$ and $F(g \circ f)$ are isomorphic (via the composition constraint $\mu _{g,f}$), but not that they are identical.
Example 2.2.4.14. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $\operatorname{\mathcal{D}}$ be an ordinary category, which we regard as a $2$-category having only identity $2$-morphisms. If $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is lax functor of $2$-categories, then its values on the $1$-morphisms of $\operatorname{\mathcal{C}}$ must satisfy the following conditions:
- $(1)$
If $u,v: X \rightarrow Y$ are $1$-morphisms of $\operatorname{\mathcal{C}}$ having the same source and target and $\gamma : u \Rightarrow v$ is a $2$-morphism of $\operatorname{\mathcal{C}}$, then $F(u) = F(v)$ (since $F(\gamma ): F(u) \Rightarrow F(v)$ must be an identity $2$-morphism of $\operatorname{\mathcal{D}}$).
- $(2)$
If $u: X \rightarrow Y$ and $v: Y \rightarrow Z$ are composable $1$-morphisms of $\operatorname{\mathcal{C}}$, then $F(v \circ u) = F(v) \circ F(u)$ (since the composition constraint $\mu _{v,u}: F(v) \circ F(u) \Rightarrow F(v \circ u)$ is an identity $2$-morphism of $\operatorname{\mathcal{D}}$).
- $(3)$
For every object $X \in \operatorname{\mathcal{C}}$, $F( \operatorname{id}_ X )$ is the identity morphism $\operatorname{id}_{ F(X)}$ in $\operatorname{\mathcal{D}}$ (since the identity constraint $\epsilon _{X}: \operatorname{id}_{ F(X)} \Rightarrow F( \operatorname{id}_ X )$ is an identity $2$-morphism of $\operatorname{\mathcal{D}}$).
Conversely, any specification of the values of $F$ on objects and $1$-morphisms which satisfies conditions $(1)$, $(2)$, and $(3)$ extends uniquely to a strict functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ (the coherence conditions appearing in Definition 2.2.4.5 are automatic, by virtue of the fact that every $2$-morphism of $\operatorname{\mathcal{D}}$ is an identity). In particular, every lax functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is automatically strict. Beware that the analogous statement is generally false if the roles of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are reversed.
Notation 2.2.4.15. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories. To supply a lax $2$-functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, one must specify not only the values of $F$ on objects, $1$-morphisms, and $2$-morphisms of $\operatorname{\mathcal{C}}$, but also the identity and composition constraints
\[ \epsilon _ X: \operatorname{id}_{F(X)} \Rightarrow F( \operatorname{id}_ X ) \quad \quad \mu _{g,f}: F(g) \circ F(f) \Rightarrow F( g \circ f ). \]
In situations where we need to consider more than one lax functor at a time, we will denote these $2$-morphisms by $\epsilon _{X}^{F}$ and $\mu _{g,f}^{F}$ (to avoid ambiguity).
Exercise 2.2.4.16. In the situation of Definition 2.2.4.5, show that we can replace $(a)$ and $(b)$ by the following alternative conditions:
For every object $X \in \operatorname{\mathcal{C}}$, the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{id}_{F(X)} \circ \operatorname{id}_{ F(X)} \ar@ {=>}[r]^-{ \upsilon _{F(X)} } \ar@ {=>}[d]^{ \epsilon _{X} \circ \epsilon _{X}} & \operatorname{id}_{ F(X) } \ar@ {=>}[dd]^{ \epsilon _{X} } \\ F( \operatorname{id}_ X) \circ F( \operatorname{id}_ X ) \ar@ {=>}[d]^{ \mu _{ \operatorname{id}_ X, \operatorname{id}_ X } } & \\ F( \operatorname{id}_ X \circ \operatorname{id}_ X ) \ar@ {=>}[r]^{ F( \upsilon _{X} )} & F(\operatorname{id}_ X) } \]
commutes (in the endomorphism category $\underline{\operatorname{End}}_{\operatorname{\mathcal{C}}}(X)$).
For every $1$-morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, the vertical compositions
\[ \operatorname{id}_{F(Y)} \circ F(f) \xRightarrow { \epsilon _{Y} \circ \operatorname{id}_{F(f)} } F( \operatorname{id}_{Y} ) \circ F(f) \xRightarrow { \mu _{ \operatorname{id}_ Y, f} } F( \operatorname{id}_ Y \circ f) \]
\[ F(f) \circ \operatorname{id}_{F(X)} \xRightarrow { \operatorname{id}_{F(f)} \circ \epsilon _ X } F(f) \circ F( \operatorname{id}_ X) \xRightarrow { \mu _{ f, \operatorname{id}_ X} } F(f \circ \operatorname{id}_ X) \]
are monomorphisms in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$.
See Proposition 2.1.5.13.
Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a (lax) functor between $2$-categories. According to Example 2.2.4.11, $F$ 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 §2.3.1, 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.
Definition 2.2.4.17. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a lax functor. We say that $F$ is unitary if, for every object $X \in \operatorname{\mathcal{C}}$, the identity constraint $\epsilon _{X}: \operatorname{id}_{ F(X)} \Rightarrow F( \operatorname{id}_ X)$ is an invertible $2$-morphism of $\operatorname{\mathcal{D}}$. We say that $F$ is strictly unitary if, for every object $X \in \operatorname{\mathcal{C}}$, we have an equality $\operatorname{id}_{F(X)} = F( \operatorname{id}_ X)$ and the identity constraint $\epsilon _{X}$ is the identity $2$-morphism from $\operatorname{id}_{F(X)}$ to itself.
Consequently, it is generally harmless to assume that a unitary lax functor of $2$-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is strictly unitary.