Lemma 4.19.2 (002W)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 4: Categories
Section 4.19: Filtered colimits
Lemma 4.19.2 (cite)

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Lemma 4.19.2. Let $\mathcal{I}$ and $\mathcal{J}$ be index categories. Assume that $\mathcal{I}$ is filtered and $\mathcal{J}$ is finite. Let $M : \mathcal{I} \times \mathcal{J} \to \textit{Sets}$ , $(i, j) \mapsto M_{i, j}$ be a diagram of diagrams of sets. In this case

$\mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{lim}}\nolimits _ j M_{i, j} = \mathop{\mathrm{lim}}\nolimits _ j \mathop{\mathrm{colim}}\nolimits _ i M_{i, j}.$

In particular, colimits over $\mathcal{I}$ commute with finite products, fibre products, and equalizers of sets.

Proof. Omitted. In fact, it is a fun exercise to prove that a category is filtered if and only if colimits over the category commute with finite limits (into the category of sets). $\square$

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