The Stacks project

Lemma 4.27.9. Let $\mathcal{C}$ be a category and let $S$ be a left multiplicative system of morphisms of $\mathcal{C}$. The localization functor $Q : \mathcal{C} \to S^{-1}\mathcal{C}$ commutes with finite colimits.

Proof. Let $\mathcal{I}$ be a finite category and let $\mathcal{I} \to \mathcal{C}$, $i \mapsto X_ i$ be a functor whose colimit exists. Then using (4.27.7.1), the fact that $Y/S$ is filtered, and Lemma 4.19.2 we have

\begin{align*} \mathop{\mathrm{Mor}}\nolimits _{S^{-1}\mathcal{C}}(Q(\mathop{\mathrm{colim}}\nolimits X_ i), Q(Y)) & = \mathop{\mathrm{colim}}\nolimits _{(s : Y \to Y') \in Y/S} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(\mathop{\mathrm{colim}}\nolimits X_ i, Y') \\ & = \mathop{\mathrm{colim}}\nolimits _{(s : Y \to Y') \in Y/S} \mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X_ i, Y') \\ & = \mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathrm{colim}}\nolimits _{(s : Y \to Y') \in Y/S} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X_ i, Y') \\ & = \mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _{S^{-1}\mathcal{C}}(Q(X_ i), Q(Y)) \end{align*}

and this isomorphism commutes with the projections from both sides to the set $\mathop{\mathrm{Mor}}\nolimits _{S^{-1}\mathcal{C}}(Q(X_ j), Q(Y))$ for each $j \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$. Thus, $Q(\mathop{\mathrm{colim}}\nolimits X_ i)$ satisfies the universal property for the colimit of the functor $i \mapsto Q(X_ i)$; hence, it is this colimit, as desired. $\square$


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