Kerodon

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

Corollary 5.6.1.9. Let $\operatorname{\mathcal{C}}$ be a category. Then:

  • Construction 5.5.1.1 determines a fully faithful functor

    \[ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set}) \rightarrow \operatorname{Cat}_{/\operatorname{\mathcal{C}}} \quad \quad \mathscr {F} \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F}, \]

    whose essential image consists of the left covering functors $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$.

  • Variant 5.5.1.2 determines a fully faithful functor

    \[ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set}) \rightarrow \operatorname{Cat}_{/\operatorname{\mathcal{C}}} \quad \quad \mathscr {F} \mapsto \int ^{\operatorname{\mathcal{C}}} \mathscr {F}, \]

    whose essential image consists of the right covering functors $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$.