A cone over a diagram F:J→C with summit or apex c∈C is a natural transformation λ:c⇒F whose domain is the constant functor at c. The components (λj:c→Fj)j∈J of the natural transformation are called the legs of the cone. Explicitly:
- The data of a cone over F:J→C with summit c is a collection of morphisms λj:c→Fj, indexed by the objects j∈J.
- A family of morphisms $(\lambda_j : c \to F_j){j \in \mathsf{J}}definesaconeoverFifandonlyif,foreachmorphismf : j \to kin\mathsf{J},thefollowingtrianglecommutesin\mathsf{C}$:
\xymatrix@=10pt{ & c \ar[dl]{\lambda_j} \ar[dr]^{\lambda_k} \ Fj \ar[rr]_{Ff} & & Fk}
Dually, a cone under F with nadir c is a natural transformation λ:F⇒c, whose legs are the components $(\lambda_j : F_j \to c){j \in \mathsf{J}}.Thenaturalityconditionassertsthat,foreachmorphismf : j \to kof\mathsf{J},thetriangle \xymatrix@=10pt{ Fj \ar[dr]{\lambda_j} \ar[rr]^{Ff} & & Fk\ar[dl]^{\lambda_k} \ & c}commutesin\mathsf{C}$.
SUGGESTION: cone under
From Category Theory in Context