nLab
cocone

Idea

A cocone under a commuting diagram is an object equipped with morphisms from each vertex of the diagram into it, such that all new diagrams arising this way commute.

A cocone which is universal is a colimit.

The dual notion is cone .

Definition

Let $C$ and $D$ be categories; we generally assume that $D$ is small. Let $f:D\to C$ be a functor (called a diagram in this situation). Then a cocone (or inductive cone) over $f$ is a a pair $(e,u)$ of an object $e\in C$ and a natural transformation $u:f\to \Delta e$ (where $\Delta e$ is the constant diagram $\Delta e:D\to C$, $x\mapsto e$, $x\in D$).

Note that a cocone in $C$ is precisely a cone in the opposite category $C^{\mathrm{op}}$.

Terminology for natural transformations can also be applied to cocones. For example, a component of a cocone is a component of the natural transformation $u$; that is, the component for each object $x$ of $D$ is the morphism $u(x):f(x)\to e$.

A morphism of cocones $(e,u)\to (e\prime ,u\prime )$ is a morphism $\gamma :e\to e\prime$ in $C$ such that $\gamma \circ u_x=u{\prime }_x$ for all objects $x$ in $D$ (symbolically $(\Delta \gamma )\circ u=u\prime$); the composition being the composition of underlying morphisms in $C$. Thus cocones form a category whose initial object if it exists is a colimit of $f$.