The Stacks project

Remark 4.27.15. Let $\mathcal{C}$ be a category. Let $S$ be a right multiplicative system. Given an object $X$ of $\mathcal{C}$ we denote $S/X$ the category whose objects are $s : X' \to X$ with $s \in S$ and whose morphisms are commutative diagrams

\[ \xymatrix{ X' \ar[rd]_ s \ar[rr]_ a & & X'' \ar[ld]^ t \\ & X } \]

where $a : X' \to X''$ is arbitrary. The category $S/X$ is cofiltered (see Definition 4.20.1). (This is dual to the corresponding statement in Remark 4.27.7.) Now the combined results of Lemmas 4.27.13 and 4.27.14 tell us that
\begin{equation} \label{categories-equation-right-localization-morphisms-colimit} \mathop{\mathrm{Mor}}\nolimits _{S^{-1}\mathcal{C}}(X, Y) = \mathop{\mathrm{colim}}\nolimits _{(s : X' \to X) \in (S/X)^{opp}} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X', Y) \end{equation}

This formula expressing morphisms in $S^{-1}\mathcal{C}$ as a filtered colimit of morphisms in $\mathcal{C}$ is occasionally useful.

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