Kerodon

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

Lemma 6.2.2.8. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Then $\operatorname{\mathcal{C}}'$ is reflective if and only if there exists a $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ and a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow L$ which exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor.

Proof. Assume that $\operatorname{\mathcal{C}}'$ is a reflective subcategory of $\operatorname{\mathcal{C}}$; we will show that there exists a functor $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ and a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow L$ which exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor (the reverse implication is immediate from the definitions). Let $\operatorname{\mathcal{E}}$ be the full subcategory of $\operatorname{\mathcal{C}}\times \Delta ^1$ spanned by those objects $(X,i)$ having the property that if $i=1$, then $X$ belongs to the full subcategory $\operatorname{\mathcal{C}}'$. Let $\pi : \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ denote the projection map. Let $\widetilde{u}: (X,0) \rightarrow (Y,1)$ be a morphism in $\operatorname{\mathcal{E}}$, corresponding to a morphism $u: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ for which the target $Y$ belongs to $\operatorname{\mathcal{C}}'$. By virtue of Corollary 5.1.2.4, the morphism $\widetilde{u}$ is $\pi$-cocartesian if and only if $u$ exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-localization of $X$. Consequently, our assumption that $\operatorname{\mathcal{C}}'$ is a reflective subcategory of $\operatorname{\mathcal{C}}$ guarantees that $\pi$ is a cocartesian fibration of $\infty$-categories. Applying Proposition 5.2.2.5, we deduce that there exists a functor

$L: \operatorname{\mathcal{C}}\simeq \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}\rightarrow \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{E}}\simeq \operatorname{\mathcal{C}}'$

and a morphism $\widetilde{\eta }: \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow L$ in the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ which carries each object $X \in \operatorname{\mathcal{C}}$ to a $\pi$-cocartesian morphism $(X,0) \rightarrow (L(X),1)$ in $\operatorname{\mathcal{E}}$. Composing with the projection map $\pi : \operatorname{\mathcal{E}}\rightarrow \Delta ^1$, we obtain a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow L$ in $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ which exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor. $\square$