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Andrew Critch asks at the 20-questions seminar:

In Set and AbGrp (the categories of sets and abelian groups, respectively), finite limits commute with directed colimits. As an example, if you're working with sheaves of sets, you can take kernels (a type of limit) and stalks (a type of directed colimit) safely.

Is there an example of a category where they don't commute?

(Depending on how you choose to talk about topoi, this condition is sometimes an axiom, capturing the idea of "looking sufficiently like sets".)

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    $\begingroup$ Already in your examples, the dual notions -- finite colimits and codirected limits don't commute, so it is enough to take the opposite category. Right ? $\endgroup$
    – Zoran Skoda
    Commented May 16, 2011 at 21:00
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    $\begingroup$ You need to get out and meet more categories! $\endgroup$
    – Paul Taylor
    Commented Dec 20, 2015 at 10:06

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Consider the poset of closed subsets of $[0,1]$. Let $a=\{0,1\}$ and $b(r)=[0,r]$ for $r<1$. Then the (directed) colimit of the $b(r)$ is $b=[0,1]$, and the product (i.e., intersection) of $b$ and a is $a$. However, the colimit of the products of the $b(r)$ with $a$ is $\{0\}$.

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Here is a non exotic example:

In the category of topological spaces, the functor $Y \mapsto X \times Y$ does not commute with directed colimits in general. However, if $X$ is locally compact it does.

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  • $\begingroup$ This is not an example: taking the product with a fixed topological spaces is not a case of taking a limit over a finite diagram of topological spaces. $\endgroup$
    – Bastiaan Cnossen
    Commented Oct 14, 2023 at 15:39

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