Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

2.2.3 Opposite and Conjugate $2$-Categories

Recall that every ordinary category $\operatorname{\mathcal{C}}$ has an opposite category $\operatorname{\mathcal{C}}^{\operatorname{op}}$, in which the objects are the same but the order of composition is reversed. In the setting of $2$-categories, this operation generalizes in two essentially different ways: we can independently reverse the order of either vertical or horizontal composition. To avoid confusion, we will use different terminology when discussing these two operations.

Construction 2.2.3.1 (The Opposite of a $2$-Category). Let $\operatorname{\mathcal{C}}$ be a $2$-category. We define a new $2$-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ as follows:

  • The objects of $\operatorname{\mathcal{C}}^{\operatorname{op}}$ are the objects of $\operatorname{\mathcal{C}}$. To avoid confusion, for each object $X \in \operatorname{\mathcal{C}}$ we will write $X^{\operatorname{op}}$ for the corresponding object of $\operatorname{\mathcal{C}}^{\operatorname{op}}$.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we have $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}^{\operatorname{op}}}( X^{\operatorname{op}}, Y^{\operatorname{op}} ) = \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y, X)$. In particular, every $1$-morphism $f: Y \rightarrow X$ in the $2$-category $\operatorname{\mathcal{C}}$ can be regarded as a $1$-morphism from $X^{\operatorname{op}}$ to $Y^{\operatorname{op}}$ in the $2$-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$, which we will denote by $f^{\operatorname{op}}: X^{\operatorname{op}} \rightarrow Y^{\operatorname{op}}$. Similarly, if we are given a pair of $1$-morphisms $f,g: Y \rightarrow X$ in the $2$-category $\operatorname{\mathcal{C}}$ having the same source and target, then every $2$-morphism $\gamma : f \Rightarrow g$ in $\operatorname{\mathcal{C}}$ determines a $2$-morphism from $f^{\operatorname{op}}$ to $g^{\operatorname{op}}$ in $\operatorname{\mathcal{C}}^{\operatorname{op}}$, which we will denote by $\gamma ^{\operatorname{op}}: f^{\operatorname{op}} \Rightarrow g^{\operatorname{op}}$.

  • For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the composition functor

    \[ \circ : \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}^{\operatorname{op}}}( Y^{\operatorname{op}}, Z^{\operatorname{op}}) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}^{\operatorname{op}}}(X^{\operatorname{op}},Y^{\operatorname{op}}) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}^{\operatorname{op}}}(X^{\operatorname{op}},Z^{\operatorname{op}}) \]

    for the $2$-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is given by the the composition functor

    \[ \circ : \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,X) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Z,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Z, X). \]

    on the $2$-category $\operatorname{\mathcal{C}}$; in particular, it is given on objects by the formula $f^{\operatorname{op}} \circ g^{\operatorname{op}} = (g \circ f)^{\operatorname{op}}$.

  • For every object $X \in \operatorname{\mathcal{C}}$, the identity $1$-morphism $\operatorname{id}_{ X^{\operatorname{op}} } \in \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}^{\operatorname{op}}}( X^{\operatorname{op}}, X^{\operatorname{op}} )$ is given by $\operatorname{id}_{X}^{\operatorname{op}}$, where $\operatorname{id}_{X} \in \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$ is the identity $1$-morphism associated to $X$ in the $2$-category $\operatorname{\mathcal{C}}$, and the unit constraint $\upsilon _{X^{\operatorname{op}}}$ is the isomorphism $\upsilon _{X}^{\operatorname{op}}: \operatorname{id}_{X^{\operatorname{op}}} \circ \operatorname{id}_{X^{\operatorname{op}}} \xRightarrow {\sim } \operatorname{id}_{X^{\operatorname{op}} }$.

  • For every triple of composable $1$-morphisms

    \[ W \xrightarrow {f} X \xrightarrow {g} Y \xrightarrow {h} Z \]

    in the $2$-category $\operatorname{\mathcal{C}}$, the associativity constraint

    \[ \alpha _{ f^{\operatorname{op}}, g^{\operatorname{op}}, h^{\operatorname{op}} }: f^{\operatorname{op}} \circ (g^{\operatorname{op}} \circ h^{\operatorname{op}} ) \xRightarrow {\sim } ( f^{\operatorname{op}} \circ g^{\operatorname{op}} ) \circ h^{\operatorname{op}} \]

    in the $2$-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is given by the inverse $(\alpha ^{\operatorname{op}}_{h,g,f})^{-1}$ of the associativity constraint $\alpha _{h,g,f}: h \circ (g \circ f) \xRightarrow {\sim } (h \circ g) \circ f$ in the $2$-category $\operatorname{\mathcal{C}}$.

We will refer to $\operatorname{\mathcal{C}}^{\operatorname{op}}$ as the opposite of the $2$-category $\operatorname{\mathcal{C}}$.

Example 2.2.3.2. Let $\operatorname{\mathcal{C}}$ be a category which admits fiber products, and let $\operatorname{Corr}(\operatorname{\mathcal{C}})$ be the $2$-category of correspondences in $\operatorname{\mathcal{C}}$ (see Example 2.2.2.1). Then the opposite $2$-category $\operatorname{Corr}(\operatorname{\mathcal{C}})^{\operatorname{op}}$ can be identified with $\operatorname{Corr}(\operatorname{\mathcal{C}})$ itself (every correspondence from $X$ to $Y$ in $\operatorname{\mathcal{C}}$ can also be viewed as a correspondence from $Y$ to $X$).

Example 2.2.3.3. Let $\operatorname{\mathcal{C}}$ be a monoidal category, and let $B\operatorname{\mathcal{C}}$ be the $2$-category obtained by delooping $\operatorname{\mathcal{C}}$ (Example 2.2.2.4). Then the opposite $2$-category $(B\operatorname{\mathcal{C}})^{\operatorname{op}}$ can be identified with $B(\operatorname{\mathcal{C}}^{\operatorname{rev}})$, where $\operatorname{\mathcal{C}}^{\operatorname{rev}}$ denotes the reverse of the monoidal category $\operatorname{\mathcal{C}}$ (Example 2.1.3.5).

Construction 2.2.3.4 (The Conjugate of a $2$-Category). Let $\operatorname{\mathcal{C}}$ be a $2$-category. We define a new $2$-category $\operatorname{\mathcal{C}}^{\operatorname{c}}$ as follows:

  • The objects of $\operatorname{\mathcal{C}}^{\operatorname{c}}$ are the objects of $\operatorname{\mathcal{C}}$. To avoid confusion, for each object $X \in \operatorname{\mathcal{C}}$ we will write $X^{\operatorname{c}}$ for the corresponding object of $\operatorname{\mathcal{C}}^{\operatorname{c}}$.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we have $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}^{\operatorname{c}}}( X^{\operatorname{c}}, Y^{\operatorname{c}} ) = \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X, Y)^{\operatorname{op}}$. In particular, every $1$-morphism $f: X \rightarrow Y$ in the $2$-category $\operatorname{\mathcal{C}}$ can be regarded as a $1$-morphism from $X^{\operatorname{c}}$ to $Y^{\operatorname{c}}$ in the $2$-category $\operatorname{\mathcal{C}}^{\operatorname{c}}$, which we will denote by $f^{\operatorname{c}}: X^{\operatorname{c}} \rightarrow Y^{\operatorname{c}}$. Similarly, if we are given a pair of $1$-morphisms $f,g: X \rightarrow Y$ in the $2$-category $\operatorname{\mathcal{C}}$ having the same source and target, then every $2$-morphism $\gamma : f \Rightarrow g$ in $\operatorname{\mathcal{C}}$ determines a $2$-morphism from $g^{\operatorname{c}}$ to $f^{\operatorname{c}}$ in $\operatorname{\mathcal{C}}^{\operatorname{c}}$, which we will denote by $\gamma ^{\operatorname{c}}: g^{\operatorname{c}} \Rightarrow f^{\operatorname{c}}$.

  • For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the composition functor

    \[ \circ : \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}^{\operatorname{c}}}( Y^{\operatorname{c}}, Z^{\operatorname{c}}) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}^{\operatorname{c}}}(X^{\operatorname{c}},Y^{\operatorname{c}}) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}^{\operatorname{op}}}(X^{\operatorname{c}},Z^{\operatorname{c}}) \]

    for the $2$-category $\operatorname{\mathcal{C}}^{\operatorname{c}}$ is induced by the composition functor

    \[ \circ : \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X, Z). \]

    on $\operatorname{\mathcal{C}}$ by passing to opposite categories. In particular, it is given on objects by the formula $g^{\operatorname{c}} \circ f^{\operatorname{c}} = (g \circ f)^{\operatorname{c}}$.

  • For every object $X \in \operatorname{\mathcal{C}}$, the identity $1$-morphism $\operatorname{id}_{ X^{\operatorname{c}} } \in \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}^{\operatorname{c}}}( X^{\operatorname{c}}, X^{\operatorname{c}} )$ is given by $\operatorname{id}_{X}^{\operatorname{c}}$, where $\operatorname{id}_{X} \in \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$ is the identity $1$-morphism associated to $X$ in the $2$-category $\operatorname{\mathcal{C}}$, and the unit constraint $\upsilon _{X^{\operatorname{c}}}$ is the isomorphism $(\upsilon _{X}^{\operatorname{c}})^{-1}: \operatorname{id}_{X^{\operatorname{c}} } \circ \operatorname{id}_{X^{\operatorname{c}} } \xRightarrow {\sim } \operatorname{id}_{ X^{\operatorname{c}} }$.

  • For every triple of composable $1$-morphisms

    \[ W \xrightarrow {f} X \xrightarrow {g} Y \xrightarrow {h} Z \]

    in the $2$-category $\operatorname{\mathcal{C}}$, the associativity constraint

    \[ \alpha _{ h^{\operatorname{c}},g^{\operatorname{c}}, f^{\operatorname{c}} }: h^{\operatorname{c}} \circ (g^{\operatorname{c}} \circ f^{\operatorname{c}} ) \xRightarrow {\sim } ( h^{\operatorname{c}} \circ g^{\operatorname{c}} ) \circ f^{\operatorname{c}} \]

    in the $2$-category $\operatorname{\mathcal{C}}^{\operatorname{c}}$ is given by the inverse $(\alpha ^{\operatorname{c}}_{h,g,f})^{-1}$ of the associativity constraint $\alpha _{h,g,f}: h \circ (g \circ f) \xRightarrow {\sim } (h \circ g) \circ f$ in the $2$-category $\operatorname{\mathcal{C}}$.

We will refer to $\operatorname{\mathcal{C}}^{\operatorname{c}}$ as the conjugate of the $2$-category $\operatorname{\mathcal{C}}$.

Example 2.2.3.5. Let $\operatorname{\mathcal{C}}$ be a monoidal category, and let $B\operatorname{\mathcal{C}}$ be the $2$-category obtained by delooping $\operatorname{\mathcal{C}}$ (Example 2.2.2.4). Then the conjugate $2$-category $(B\operatorname{\mathcal{C}})^{\operatorname{c}}$ can be identified with $B(\operatorname{\mathcal{C}}^{\operatorname{op}})$, where we endow the opposite category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ with the monoidal structure of Example 2.1.3.4.

Remark 2.2.3.6. Constructions 2.2.3.1 and 2.2.3.4 are analogous but not identical. At the level of $2$-morphisms, passage from a $2$-category $\operatorname{\mathcal{C}}$ to its opposite $\operatorname{\mathcal{C}}^{\operatorname{op}}$ reverses the order of horizontal composition, but preserves the order of vertical composition; passage from $\operatorname{\mathcal{C}}$ to its conjugate $\operatorname{\mathcal{C}}^{\operatorname{c}}$ preserves the order of horizontal composition and reverses the order of vertical composition. Following the notation of Warning 2.2.1.8, we have

\[ \delta ^{\operatorname{op}} \gamma ^{\operatorname{op}} = ( \delta \gamma )^{\operatorname{op}} \quad \quad \gamma ^{\operatorname{op}} \circ \gamma '^{\operatorname{op}} = (\gamma ' \circ \gamma )^{\operatorname{op}} \]
\[ \gamma ^{\operatorname{c}} \delta ^{\operatorname{c}} = ( \delta \gamma )^{\operatorname{c}} \quad \quad \gamma '^{\operatorname{c}} \circ \gamma ^{\operatorname{c}} = (\gamma ' \circ \gamma )^{\operatorname{c}}. \]

Example 2.2.3.7. Let $\operatorname{\mathcal{C}}$ be an ordinary category, which we regard as a $2$-category having only identity $2$-morphisms (Example 2.2.0.6). Then the opposite $2$-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ of Construction 2.2.3.1 coincides with the opposite of $\operatorname{\mathcal{C}}$ as an ordinary category (which we can again regard as a $2$-category having only identity morphisms). The conjugate $2$-category $\operatorname{\mathcal{C}}^{\operatorname{c}}$ of Construction 2.2.3.4 can be identified with $\operatorname{\mathcal{C}}$ itself.