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

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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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 ? –  Zoran Skoda May 16 '11 at 21:00

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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