Colimit ($\mathrm{colim}$)

A colimit is a universal cocone under a diagram: every other cocone factors through it uniquely.

Limits of shape $\mathcal{I}$ commute with colimits of shape $\mathcal{J}$ when, for every $D:\mathcal{I}\times \mathcal{J}\to \mathcal{C}$, the canonical comparisonis an isomorphism whenever both sides exist.

A category is filtered when every finite diagram in it admits a cocone. Equivalently, it is nonempty, every pair of objects maps to a common object, and every pair of parallel arrows becomes equal after postcomposition.

A category is weakly filtered when every finite connected diagram in it admits a cocone. Equivalently, each of its connected components is a filtered category.

In the Category of sets, filtered colimits commute with finite limits. Elements in a finite limiting diagram involve only finitely many representatives and finitely many equalities, all of which can be realized at one common stage of a filtered diagram.

A local state classifier of $\mathcal{C}$ is a colimit of the inclusion of the wide subcategory containing all objects and only monomorphisms into $\mathcal{C}$.