## Tag `09WQ`

Chapter 4: Categories > Section 4.19: Filtered colimits

Lemma 4.19.4. Let $\mathcal{I}$ be an index category, i.e., a category. Assume that for every pair of objects $x, y$ of $\mathcal{I}$ there exists an object $z$ and morphisms $x \to z$ and $y \to z$. Then

- If $M$ and $N$ are diagrams of sets over $\mathcal{I}$, then $\mathop{\rm colim}\nolimits (M_i \times N_i) \to \mathop{\rm colim}\nolimits M_i \times \mathop{\rm colim}\nolimits N_i$ is surjective,
- in general colimits of diagrams of sets over $\mathcal{I}$ do not commute with finite nonempty products.

Proof.Proof of (1). Let $(\overline{m}, \overline{n})$ be an element of $\mathop{\rm colim}\nolimits M_i \times \mathop{\rm colim}\nolimits N_i$. Then we can find $m \in M_x$ and $n \in N_y$ for some $x, y \in \mathop{\rm Ob}\nolimits(\mathcal{I})$ such that $m$ mapsto $\overline{m}$ and $n$ mapsto $\overline{n}$. See Section 4.15. Choose $a : x \to z$ and $b : y \to z$ in $\mathcal{I}$. Then $(M(a)(m), N(b)(n))$ is an element of $(M \times N)_z$ whose image in $\mathop{\rm colim}\nolimits (M_i \times N_i)$ maps to $(\overline{m}, \overline{n})$ as desired.Proof of (2). Let $G$ be a non-trivial group and let $\mathcal{I}$ be the one-object category with endomorphism monoid $G$. Then $\mathcal{I}$ trivially satisfies the condition stated in the lemma. Now let $G$ act on itself by translation and view the $G$-set $G$ as a set-valued $\mathcal{I}$-diagram. Then $$ \mathop{\rm colim}\nolimits_\mathcal{I} G \times \mathop{\rm colim}\nolimits_\mathcal{I} G \cong G/G \times G/G $$ is not isomorphic to $$ \mathop{\rm colim}\nolimits_\mathcal{I} (G \times G) \cong (G \times G)/G $$ This example indicates that you cannot just drop the additional condition Lemma 4.19.2 even if you only care about finite products. $\square$

The code snippet corresponding to this tag is a part of the file `categories.tex` and is located in lines 2088–2101 (see updates for more information).

```
\begin{lemma}
\label{lemma-preserve-products}
Let $\mathcal{I}$ be an index category, i.e., a category. Assume
that for every pair of objects $x, y$ of $\mathcal{I}$
there exists an object $z$ and morphisms $x \to z$ and $y \to z$.
Then
\begin{enumerate}
\item If $M$ and $N$ are diagrams of sets over $\mathcal{I}$,
then $\colim (M_i \times N_i) \to \colim M_i \times \colim N_i$
is surjective,
\item in general colimits of diagrams of sets over $\mathcal{I}$
do not commute with finite nonempty products.
\end{enumerate}
\end{lemma}
\begin{proof}
Proof of (1). Let $(\overline{m}, \overline{n})$
be an element of $\colim M_i \times \colim N_i$.
Then we can find $m \in M_x$ and $n \in N_y$ for some
$x, y \in \Ob(\mathcal{I})$ such that $m$ mapsto
$\overline{m}$ and $n$ mapsto $\overline{n}$. See
Section \ref{section-limit-sets}.
Choose $a : x \to z$ and $b : y \to z$
in $\mathcal{I}$. Then $(M(a)(m), N(b)(n))$ is an element of
$(M \times N)_z$ whose image in $\colim (M_i \times N_i)$
maps to $(\overline{m}, \overline{n})$ as desired.
\medskip\noindent
Proof of (2). Let $G$ be a non-trivial group and
let $\mathcal{I}$ be the one-object category with endomorphism monoid $G$.
Then $\mathcal{I}$ trivially satisfies the condition stated in the lemma.
Now let $G$ act on itself by translation and view the $G$-set $G$
as a set-valued $\mathcal{I}$-diagram. Then
$$
\colim_\mathcal{I} G \times \colim_\mathcal{I} G \cong G/G \times G/G
$$
is not isomorphic to
$$
\colim_\mathcal{I} (G \times G) \cong (G \times G)/G
$$
This example indicates that you cannot just drop the additional
condition Lemma \ref{lemma-directed-commutes}
even if you only care about finite products.
\end{proof}
```

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