Definition:Cone (Category Theory)

From ProofWiki
Jump to navigation Jump to search

This page is about Cone in the context of Category Theory. For other uses, see Cone.


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 $\mathbf J$:

$\begin{xy}\xymatrix@+0.5em@L+2px{ C \ar[d]_*+{c_i} \ar[dr]^*+{c_j} \\ D_i \ar[r]_*+{D_\alpha} & D_j }\end{xy}$

is a commutative diagram.

Also see

Linguistic Note

The adjectival form of cone is conical.