On Fri, 30 Jul 2004, Tom Leinster wrote:
I've recently come across the following curious little result. I know how to prove it and have a use for it, but my question is: can anyone supply a wider context or explanation?
The result is that the limit in Set of any diagram
... ---> S_3 ---> S_2 ---> S_1
of finite nonempty sets is nonempty. Note that finiteness cannot be dropped: for instance, take each S_n to be the natural numbers and each map to be addition of 1.
I'm not sure whether this counts as an explanation, but it's certainly a wider context: the result is a special case of the fact that a (cofiltered) inverse limit of locales and proper maps maps properly to each of the vertices of the diagram. See C3.2.11 in the Elephant, and also C1.1.12. (And note that the result fails for spaces: thus a key part of the argument in the finite case is that the inverse limit in Loc is a spatial locale.) Peter Johnstone