Kerodon

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Variant 9.1.1.22. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which is colocalizing, and let $\operatorname{\mathcal{C}}'$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-colocal objects. Then:

$(1)$

The full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is coreflective.

$(2)$

The inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ admits a right adjoint $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$.

$(3)$

A morphism $w$ of $\operatorname{\mathcal{C}}$ is contained in $W$ if and only if $L(w)$ is an isomorphism in $\operatorname{\mathcal{C}}'$.

$(4)$

The functor $L$ exhibits $\operatorname{\mathcal{C}}'$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.