category theory Archives

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 …

Thuses

Fun example: Empty colimit does not commute with empty limit