# Definition:Cone (Category Theory)

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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 **cone to $D$** comprises an object $C$ of $\mathbf C$, and a morphism:

- $c_j: C \to D_j$

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

is a commutative diagram.

## Also see

- Cocone, the dual notion.

## Sources

- 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 5.4$: Definition $5.15$