Kerodon

Remark 6.3.0.2 (Existence and Uniqueness). Let $\operatorname{\mathcal{C}}$ be a category and let $W$ be a collection of morphisms in $\operatorname{\mathcal{C}}$. Then there exists a category $W^{-1} \operatorname{\mathcal{C}}$ and a functor $F: \operatorname{\mathcal{C}}\rightarrow W^{-1} \operatorname{\mathcal{C}}$ which exhibits $W^{-1} \operatorname{\mathcal{C}}$ as a strict localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Moreover, the category $W^{-1} \operatorname{\mathcal{C}}$ is determined uniquely up to isomorphism. In what follows, we will sometimes abuse terminology by referring to $W^{-1} \operatorname{\mathcal{C}}$ as the strict localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Explicitly, the category $W^{-1} \operatorname{\mathcal{C}}$ can be constructed from $\operatorname{\mathcal{C}}$ by adjoining a new morphism $w^{-1}: Y \rightarrow X$ for each morphism $w: X \rightarrow Y$ of $W$, and imposing the relations $w^{-1} \circ w = \operatorname{id}_{X}$ and $w \circ w^{-1} = \operatorname{id}_{Y}$. From this description, we see that the functor $F$ induces a bijection $\operatorname{Ob}(\operatorname{\mathcal{C}}) \simeq \operatorname{Ob}(W^{-1} \operatorname{\mathcal{C}})$.