Fun example: Empty colimit does not commute with empty limit
One important property 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 used for example to check that a continuous morphism of sites commuting with finite limits induces a morphism of topoi , i.e. the pullback map is exact (note that a continuous morphism of sites does not induce a morphism of topoi in the general case).
The definition of a filtered system has a somewhat strange condition in that it is non-empty. For a while I was not sure why it is essential to require this property. It turns out that without this condition, 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 set .
Similarly, the 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!
A comment from this paper might be relevant. They describe closed classes of colimits.
Three more classes arise from the anomalous behavior of the empty set (the empty colimit does not commute with the empty limit, but does commute with all nonempty ones): we may add the empty colimit to the class of filtered colimits the class of sifted colimits, to obtain the classes which commute with all nonempty finite limits (resp. with nonempty finite products), and by cutting down the class of all colimits to the class of connected colimits, we obtain the class which commutes (with all conical limits and) with the empty limit.
One way (e.g. as found on nlab) to define a filtered category avoids a special mention of pointedness: a category is filtered precisely when every finite diagram has a co-cone. This feels to me analogous to how one may define a topological space using two axioms (closed under arbitrary union and finitary intersection) instead of the popular four axioms (closed under nullary union and arbitrary non-nullary union; closed under nullary intersection and binary intersection). Another similar shift in viewpoint is the passage from standard natural induction [[if P(0) and if for each n, P(n) implies P(n+1), then P holds everywhere]] to `strong natural induction’ [[if for each n (P(m) is true for each m<n) implies P(n), then P holds everywhere]].