# Definition:Cocone

## Definition

Let $\mathbf C$ be a metacategory.

Let $D: \mathbf J \to \mathbf C$ be a $\mathbf J$-diagram in $\mathbf C$.

A cocone from $D$ comprises an object $C$ of $\mathbf C$, and a morphism:

$c_j: D_j \to C$

for each object of $\mathbf J$, such that for each morphism $\alpha: i \to j$ of $\mathbf J$:

$\begin{xy}\[email protected][email protected]+2px{ D_i \ar[r]^*+{D_\alpha} \ar[dr]_*+{c_i} & D_j \ar[d]^*+{c_j} \\ & C }\end{xy}$

is a commutative diagram.

## Also known as

Some authors, notably Saunders Mac Lane, dislike the name cocone and rather speak of cones from the base $D$.

Cones are then called cones to the base $D$.

So as to avoid the unavoidable ambiguity this gives rise to, on this web site, cocone is the designated term.