TITLE: Is there a category in which finite limits and directed colimits *don't* commute
QUESTION [7 upvotes]: 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".)

REPLY [5 votes]: 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.