# Kerodon

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Construction 2.2.8.9 (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 $\operatorname{\mathcal{C}}$.