One of the important properties of filtered colimits is that they commute with finite limits in the category of sets.
Theorem: Let be a functor, where is a filtered small category and is a finite category. Then the natural mapping
is an isomorphism.
This statement is, for example, useful to check that a continuous morphism of sites commuting with finite limits induces a topos morphism , i.e. the pullback map is exact (note that a continuous site morphism does not induce a topos morphism in the general case).
The definition of a filtered system has a somewhat strange condition that it is non-empty. For some time I was not sure why it is essential to require this property. It turns out that without this condition, the filtered colimits will not commute with finite limits. Namely, the empty colimit will not commute with the empty limit (and only with it!).
I claim that the limit over an empty diagram in any category is simply a final object in that category.
Indeed, let be an empty diagram in a category . Then is an object in such that any other object admits exactly one morphism . This means that is a final object in . Clearly, a final object in is the one-point sets .
Similarly, colimit on an empty diagram is nothing more than an initial object in . Clearly, an initial object in is the empty set .
Then we apply these considerations to the empty diagram , the natural map
is the map
that is clearly not an isomorphism!