Remark 6.3.2.2 (01N1): Uniqueness of Localizations

Remark 6.3.2.2 (Uniqueness of Localizations). Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$. Proposition 6.3.2.1 asserts that there exists an $\infty $-category $\operatorname{\mathcal{D}}$ and a morphism $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. In this case, for every $\infty $-category $\operatorname{\mathcal{E}}$, composition with $F$ induces a bijection

\[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) = \pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq } ) \]

(Proposition 6.3.1.13). In other words, the $\infty $-category $\operatorname{\mathcal{D}}$ corepresents the functor

\[ \mathrm{h} \mathit{\operatorname{Cat}_{\infty }} \rightarrow \operatorname{Set}\quad \quad \operatorname{\mathcal{E}}\mapsto \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq } ). \]

It follows that $\operatorname{\mathcal{D}}$ is uniquely determined (up to canonical isomorphism) as an object of the homotopy category $\mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$. We will sometimes emphasize this uniqueness by referring to $\operatorname{\mathcal{D}}$ as the localization of $\operatorname{\mathcal{C}}$ with respect to $W$, and denoting it by $\operatorname{\mathcal{C}}[W^{-1}]$. Beware that the localization $\operatorname{\mathcal{C}}[W^{-1}]$ is not well-defined up to isomorphism as a simplicial set: in fact, any equivalent $\infty $-category can also be regarded as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$ (Remark 6.3.1.19).