Kerodon

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

Construction 1.2.4.4. 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 $\operatorname{\mathcal{C}}$.