# Definition:Cocone

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## 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 $\mathbf J$:

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

is a commutative diagram.

This article is complete as far as it goes, but it could do with expansion.In particular: when time comes, add def "cocone is natural trafo $D \to \kappa_C$ with $\kappa_C$ constant functor"You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Expand}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## 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.

## Also see

## Sources

- 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 5.6$