The Stacks project

Remark 4.27.7. Let $\mathcal{C}$ be a category. Let $S$ be a left multiplicative system. Given an object $Y$ of $\mathcal{C}$ we denote $Y/S$ the category whose objects are $s : Y \to Y'$ with $s \in S$ and whose morphisms are commutative diagrams

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

where $a : Y' \to Y''$ is arbitrary. We claim that the category $Y/S$ is filtered (see Definition 4.19.1). Namely, LMS1 implies that $\text{id}_ Y : Y \to Y$ is in $Y/S$; hence $Y/S$ is nonempty. LMS2 implies that given $s_1 : Y \to Y_1$ and $s_2 : Y \to Y_2$ we can find a diagram

\[ \xymatrix{ Y \ar[d]_{s_1} \ar[r]_{s_2} & Y_2 \ar[d]^ t \\ Y_1 \ar[r]^ a & Y_3 } \]

with $t \in S$. Hence $s_1 : Y \to Y_1$ and $s_2 : Y \to Y_2$ both have maps to $t \circ s_2 : Y \to Y_3$ in $Y/S$. Finally, given two morphisms $a, b$ from $s_1 : Y \to Y_1$ to $s_2 : Y \to Y_2$ in $Y/S$ we see that $a \circ s_1 = b \circ s_1$; hence by LMS3 there exists a $t : Y_2 \to Y_3$ in $S$ such that $t \circ a = t \circ b$. Now the combined results of Lemmas 4.27.5 and 4.27.6 tell us that

4.27.7.1
\begin{equation} \label{categories-equation-left-localization-morphisms-colimit} \mathop{\mathrm{Mor}}\nolimits _{S^{-1}\mathcal{C}}(X, Y) = \mathop{\mathrm{colim}}\nolimits _{(s : Y \to Y') \in Y/S} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, Y') \end{equation}

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


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