Kerodon

Example 2.2.8.15. Let $\operatorname{\mathcal{C}}$ be a $(2,1)$-category, so that $\operatorname{Pith}(\operatorname{\mathcal{C}}) = \operatorname{\mathcal{C}}$. In particular, the inclusion $\operatorname{Pith}(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{\mathcal{C}}$ induces an isomorphism of categories $\operatorname{hPith}(\operatorname{\mathcal{C}}) \simeq \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. In this situation, we will generally abuse notation by identifying $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ with $\operatorname{hPith}(\operatorname{\mathcal{C}})$ and referring to it as the homotopy category of $\operatorname{\mathcal{C}}$.