Kerodon

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

Construction 2.3.2.10 (The Pith of a $2$-Category). Let $\operatorname{\mathcal{C}}$ be a $2$-category. We define a new $2$-category $\operatorname{Pith}(\operatorname{\mathcal{C}})$ as follows:

  • The objects of $\operatorname{Pith}(\operatorname{\mathcal{C}})$ are the objects of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the category $\underline{\operatorname{Hom}}_{\operatorname{Pith}(\operatorname{\mathcal{C}})}( X, Y)$ is the core $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)^{\simeq }$ of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ (see Construction 1.2.4.4).

  • The composition law, associativity constraints, and unit constraints of $\operatorname{Pith}(\operatorname{\mathcal{C}})$ are given by restricting the composition law, associativity constraints, and unit constraints of $\operatorname{\mathcal{C}}$.

Then $\operatorname{Pith}(\operatorname{\mathcal{C}})$ is a $(2,1)$-category which we will refer to as the pith of $\operatorname{\mathcal{C}}$.