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Tag 002W

Chapter 4: Categories > Section 4.19: Filtered colimits

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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