# The Stacks Project

## Tag 002W

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 $$\mathop{\rm colim}\nolimits_i \mathop{\rm lim}\nolimits_j M_{i, j} = \mathop{\rm lim}\nolimits_j \mathop{\rm colim}\nolimits_i M_{i, j}.$$ 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). $\square$

The code snippet corresponding to this tag is a part of the file categories.tex and is located in lines 2022–2036 (see updates for more information).

\begin{lemma}
\label{lemma-directed-commutes}
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
$$\colim_i \lim_j M_{i, j} = \lim_j \colim_i M_{i, j}.$$
In particular, colimits over $\mathcal{I}$ commute with finite products,
fibre products, and equalizers of sets.
\end{lemma}

\begin{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).
\end{proof}

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