Kerodon

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

Construction 2.2.6.8. Let $\operatorname{\mathcal{C}}$ be a $2$-category equipped with a twisting cochain

\[ \{ \mu _{g,f} \} = \{ \mu _{g,f}: (g \circ f) \Rightarrow (g \circ ' f) \} . \]

We define a new $2$-category $\operatorname{\mathcal{C}}'$ as follows:

  • The objects of $\operatorname{\mathcal{C}}'$ are the objects of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we define $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X,Y)$ to be the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$. In particular, we can identify $1$-morphisms of $\operatorname{\mathcal{C}}'$ with $1$-morphisms of $\operatorname{\mathcal{C}}$, $2$-morphisms of $\operatorname{\mathcal{C}}'$ with $2$-morphisms of $\operatorname{\mathcal{C}}$, and the vertical composition of $2$-morphisms in $\operatorname{\mathcal{C}}'$ with the vertical composition of $2$-morphisms in $\operatorname{\mathcal{C}}$.

  • For every object $X \in \operatorname{\mathcal{C}}$, the identity $1$-morphism from $X$ to itself in the $2$-category $\operatorname{\mathcal{C}}'$ is the same as the identity morphism from $X$ to itself in the $2$-category $\operatorname{\mathcal{C}}$.

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

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

    is given on objects by $(g,f) \mapsto g \circ ' f$ and on morphisms by the construction

    \[ ( \delta : g \Rightarrow g', \gamma : f \Rightarrow f') \mapsto \mu _{g',f'} (\delta \circ \gamma ) \mu _{g,f}^{-1}. \]
  • For every object $X \in \operatorname{\mathcal{C}}$, the unit constraint $\upsilon '_{X}: \operatorname{id}_{X} \circ ' \operatorname{id}_{X} \xRightarrow {\sim } \operatorname{id}_{X}$ for the $2$-category $\operatorname{\mathcal{C}}'$ is given by the composition

    \[ \operatorname{id}_{X} \circ ' \operatorname{id}_{X} \xRightarrow { \mu _{ \operatorname{id}_ X, \operatorname{id}_ X}^{-1} } \operatorname{id}_ X \circ \operatorname{id}_ X \xRightarrow { \upsilon _ X } \operatorname{id}_ X. \]
  • For every triple of composable $1$-morphisms $W \xrightarrow {f} X \xrightarrow {g} Y \xrightarrow {h} Z$ of $\operatorname{\mathcal{C}}$, the associativity constraint of $\operatorname{\mathcal{C}}'$ is given by the composition

    \begin{eqnarray*} h \circ ' (g \circ ' f) & \xRightarrow { \mu _{h, g \circ ' f}^{-1} } & h \circ (g \circ ' f) \\ & \xRightarrow { \operatorname{id}_ h \circ \mu _{g,f}^{-1}} & h \circ (g \circ f) \\ & \xRightarrow { \alpha _{h,g,f} } & (h \circ g) \circ f \\ & \xRightarrow { \mu _{h,g} \circ \operatorname{id}_ f} & (h \circ ' g) \circ f \\ & \xRightarrow { \mu _{h \circ ' g, f} } & (h \circ ' g) \circ ' f. \end{eqnarray*}

We will refer to $\operatorname{\mathcal{C}}'$ as the twist of $\operatorname{\mathcal{C}}$ with respect to $\{ \mu _{g,f} \} $.