Construction 1.3.5.4. Let $\operatorname{\mathcal{C}}$ be a category. We define a subcategory $\operatorname{\mathcal{C}}^{\simeq } \subseteq \operatorname{\mathcal{C}}$ as follows:
Every object of $\operatorname{\mathcal{C}}$ belongs to $\operatorname{\mathcal{C}}^{\simeq }$.
A morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ belongs to $\operatorname{\mathcal{C}}^{\simeq }$ if and only if $f$ is an isomorphism.
We will refer to $\operatorname{\mathcal{C}}^{\simeq }$ as the core of $\operatorname{\mathcal{C}}$.