Kerodon

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

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 $\operatorname{\mathcal{C}}$.