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2.1.4 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 2.1.4.1 (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 $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$, an $F(\gamma )$ 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

\[ \xymatrix { \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \ar [r]^-{\circ } \ar [d]^{F_{Y,Z} \times F_{X,Y}} & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d]^{F_{X,Z}} \\ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}(Y,Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}(X,Y) \ar [r]^-{ \circ } & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( X, Z ) } \]

is strictly commutative.

$(3)$

For every $1$-morphism $f: X \rightarrow Y$ in the $2$-category $\operatorname{\mathcal{C}}$, we have identities

\[ F( \lambda _{f} ) = \lambda _{F(f)} \quad \quad F( \rho _ f ) = \rho _{F(f)}. \]

In other words, $F$ carries left and right unit constraints of $\operatorname{\mathcal{C}}$ to left and right unit constraints of $\operatorname{\mathcal{D}}$.

$(4)$

For every compostable 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 $\operatorname{\mathcal{D}}$.

Remark 2.1.4.2. In the situation of Definition 2.1.4.1, conditions $(3)$ and $(4)$ are automatically satisfied if the $2$-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are strict.

Note that axiom $(2)$ of Definition 2.1.4.1 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 better to allow a more liberal notion of functor, which is only required to preserve composition up to isomorphism.

Definition 2.1.4.3 (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 compostable 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

    \[ \xymatrix@C =40pt@R=40pt{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \ar [r]^-{\circ } \ar [d]_{F_{Y,Z} \times F_{X,Y}} & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d]^{F_{X,Z}} \\ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}(Y,Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}(X,Y) \ar [r]^-{ \circ } \ar@ {=>}[]+<30pt,15pt>;+<100pt,45pt>_-{\mu } & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( X, Z ) } \]

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 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 2.1.4.4. The terminology of Definition 2.1.4.3 is not standard. In [MR0220789], BĂ©nabou 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 2.1.4.3) as weak functors or pseudofunctors (to avoid confusion with the notion of strict functor).

Example 2.1.4.5. 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 2.1.4.1). Then we can regard $F$ as a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ (in the sense of Definition 2.1.4.3) 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 2.1.4.3 reduce to conditions $(3)$ and $(4)$ of Definition 2.1.4.1). 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.

Warning 2.1.4.6. 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 2.1.4.3 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 2.1.4.7. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $\operatorname{\mathcal{D}}$ be an ordinary category. Then we can regard $\operatorname{\mathcal{D}}$ as a $2$-category having only identity $2$-morphisms (Example 2.1.1.7). It follows that every lax functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is automatically a strict functor. Beware that the analogous statement is generally false if the roles of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are reversed.

Notation 2.1.4.8. 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).

For our purposes, it will be useful to consider an intermediate notion of strictness.

Definition 2.1.4.9. 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.

Remark 2.1.4.10. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories. Every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is unitary when viewed as a lax functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$. Every strict functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is strictly unitary when viewed as a lax functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.

Remark 2.1.4.11. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be strictly unitary $2$-categories (Variant 2.1.1.5). Then a strictly unitary lax functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is given by the following data:

  • For each 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

    \[ F_{X,Y}: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ). \]
  • For every pair of composable morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ in $\operatorname{\mathcal{C}}$, a composition constraint $\mu _{g,f}: F(g) \circ F(f) \Rightarrow F(g \circ f)$, depending functorially on $f$ and $g$.

This data must be required to satisfy axiom $(c)$ of Definition 2.1.4.3, together with the identities $F( \operatorname{id}_ X) = \operatorname{id}_{F(X)}$ for each object $X \in \operatorname{\mathcal{C}}$ and $\mu _{\operatorname{id}_ Y, f} = \operatorname{id}_{ F(f)} = \mu _{f, \operatorname{id}_ X}$ for each $1$-morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$.

Remark 2.1.4.12. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a unitary lax functor. Then one can modify $F$ to produce a strictly unitary lax functor $F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ by the following explicit procedure:

  • For every object $X \in \operatorname{\mathcal{C}}$, we set $F'(X) = F(X)$.

  • For every $1$-morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ which is not an identity morphism, we set $F'(f) = F(f)$; if $X = Y$ and $f = \operatorname{id}_{X}$ we instead set $F'(f) = \operatorname{id}_{F(X)}$. In either case, we have an invertible $2$-morphism $\varphi _{f}: F'(f) \xRightarrow {\sim } F(f)$, given by

    \[ \varphi _{f} = \begin{cases} \epsilon _{X}^{F} & \text{ if } f = \operatorname{id}_ X \\ \operatorname{id}_{ F(f) } & \text{ otherwise. } \end{cases} \]
  • Let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, and let $\gamma : f \Rightarrow g$ be a $2$-morphism between $1$-morphisms $f,g: X \rightarrow Y$. We define $F'(\gamma )$ to be the vertical composition $\varphi _{g}^{-1} F(\gamma ) \varphi _{f}$.

  • For every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ in the $2$-category $\operatorname{\mathcal{D}}$, we define the composition constraint $\mu ^{F'}_{g,f}: F'(g) \circ F'(f) \Rightarrow F'(g \circ f)$ to be the vertical composition

    \[ F'(g) \circ F'(f) \xRightarrow { \varphi _{g} \circ \varphi _{f} } F(g) \circ F(f) \xRightarrow { \mu _{g,f}^{F}} F(g \circ f) \xRightarrow { \varphi _{g \circ f}^{-1} } F'( g \circ f ). \]

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.