Proposition 6.2.3.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(a)$
Let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects. Then $\operatorname{\mathcal{C}}'$ is a reflective subcategory of $\operatorname{\mathcal{C}}$ and $W$ is the collection of $\operatorname{\mathcal{C}}'$-local equivalences.
- $(b)$
There exists a reflective subcategory $\operatorname{\mathcal{C}}'' \subseteq \operatorname{\mathcal{C}}$ such that $W$ is the collection of $\operatorname{\mathcal{C}}''$-local equivalences.
- $(c)$
The collection of morphisms $W$ is localizing (Definition 6.2.3.9).
- $(d)$
The collection of morphisms $W$ satisfies conditions $(1)$, $(2_{-})$, and $(3)$ of Lemma 6.2.3.11.