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 \(F \colon \mathcal{C}\times \mathcal{D} \to \mathbf{Sets}\) be a functor, where \(\mathcal{C}\) is a filtered small category and \(\mathcal{D}\) is a finite category. Then the natural mapping

\[\mathrm{colim}_{\mathcal{C}} \lim_{\mathcal{D}} F (c, d) \to \lim_{\mathcal{D}} \mathrm{colim}_{\mathcal{C}} F(c, d)\]

is an isomorphism.

This statement is used for example to check that a continuous morphism of sites \(\mathcal{D} \to \mathcal{C}\) commuting with finite limits induces a morphism of topoi \(\mathrm{Shv}(\mathcal{C}) \to \mathrm{Shv}(\mathcal{D})\), i.e. the pullback map \(f^{-1}\) 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 \(F\colon \emptyset \to X\) be an empty diagram in a category \(X\). Then \(\lim_{\emptyset} F\) is an object in \(X\) such that any other object \(A\in X\) admits exactly one morphism \(A \to \lim_{\emptyset} F\). This means that \(\lim_{\emptyset} F\) is a final object in \(X\). Clearly, a final object in \(\mathbf{Sets}\) is the one-point set \(\{*\}\).

Similarly, the colimit on an empty diagram \(\mathrm{colim}_{\emptyset} F\) is nothing more than an initial object in \(X\). Clearly, an initial object in \(\mathbf{Sets}\) is the empty set \(\emptyset\).

Then we apply these considerations to the empty diagram \(F\colon \emptyset \to \mathbf{Sets}\), the natural map

\[\textrm{colim} \lim F\times F \to \lim \textrm{colim} F\times F\]

is the map

\[\emptyset \to \{*\}\]

that is clearly not an isomorphism!

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Slava Naprienko
3 years ago

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.

Last edited 3 years ago by Slava Naprienko
3 years ago

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]].

Last edited 3 years ago by moomitten