Is there a category in which finite limits and directed colimits *don't* commute

Question

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".)

Answer 1

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 $\{0\}$.

Answer 2

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.

Answer 3

I'm hoping that I understand directed colimits and finite limits correctly. If I don't, I'll happily delete this answer.

Here's an example that works for several categories: for example, Hausdorff topological spaces and (smooth) manifolds. Actually, in each we take the slice category of objects over the real line. Consider the coequaliser diagram of the two obvious inclusions ℝ∖{0} → ℝ⨿ℝ; each object here is viewed as a manifold over ℝ in the obvious way. The coequaliser of this diagram in the category of manifolds over ℝ is ℝ (it ought to be ℝ with a double point at 0, but then we take the Hausdorffification of that to get the real line itself). Now consider the product in the slice category of this diagram with the manifold {0}, viewed as a manifold over ℝ with the obvious inclusion. The product with ℝ∖{0} is the empty set and the product with ℝ⨿ℝ is {0}⨿{0} so the coequaliser is just {0}⨿{0} again, but the product with ℝ is just {0}.

Note that this doesn't go away if you work with non-Hausdorff objects, so it would still work if we'd started with either topological spaces or non-Hausdorff manifolds. However, the difference isn't quite so stark for these so it's not so good an example. What you get in these cases is that the product with the coequaliser is {0}⨿{0} with the indiscrete topology and the coequaliser of the products is {0}⨿{0} with the discrete topology.

(Full disclosure: this example is based on one taken from the page on Frölicher spaces on the n-lab which was written by ... er ... me. There it illustrates the fact that the category of Frölicher spaces is not locally cartesian closed. Note that this is one of the requirements for a topos. The fact that this example works for manifolds shows that there is no topos in which the category of smooth manifolds embeds in such a way as to preserve both limits and colimits. But that's getting off-topic.)