MathGloss

A cone over a diagram $F : \mathsf{J} \to \mathsf{C}$ with summit or apex $c \in \mathsf{C}$ is a natural transformation $\lambda : c \Rightarrow F$ whose domain is the constant functor at $c$. The components $(\lambda_j : c \to Fj)_{j\in \mathsf{J}}$ of the natural transformation are called the legs of the cone. Explicitly:

• The data of a cone over $F : \mathsf{J} \to \mathsf{C}$ with summit $c$ is a collection of morphisms $\lambda_j : c \to Fj$, indexed by the objects $j \in \mathsf{J}$.
• A family of morphisms $(\lambda_j : c \to F_j){j \in \mathsf{J}}$defines a cone over$F$if and only if, for each morphism$f : j \to k$in$\mathsf{J}$, the following triangle commutes in$\mathsf{C}$: \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 $\lambda : F \Rightarrow c$, whose legs are the components $(\lambda_j : F_j \to c){j \in \mathsf{J}}$. The naturality condition asserts that, for each morphism$f : j \to k$of$\mathsf{J}$, the triangle$ \xymatrix@=10pt{ Fj \ar[dr]{\lambda_j} \ar[rr]^{Ff} & & Fk\ar[dl]^{\lambda_k} \ & c}$commutes in$\mathsf{C}$.

SUGGESTION: cone under

From Category Theory in Context

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