nLab cocone

Idea

A cocone under a 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 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 $\Delta e:D\to C$, $x\mapsto e$, $x\in D$). In other words, a diagram as follows, together with a natural transformation going south west to north east.

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

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 $(e,u)\to (e',u')$ is a morphism $\gamma:e\to e'$ in $C$ such that $\gamma\circ u_x=u'_x$ for all objects $x$ in $D$ (symbolically $(\Delta \gamma)\circ u = u'$); 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 $f$.