Cone (Category Theory)

For a diagram \(F: J \to C\), a cone over \(F\) with summit (or apex) \(c \in C\) is a natural transformation \(\lambda: \Delta c \Rightarrow F\), where \(\Delta c\) is the constant functor at \(c\).

The Legs of a Cone

The components \(\lambda_j: c \to Fj\) (for \(j \in J\)) of a cone are called the legs of the cone.

Cocones

A natural transformation \(\lambda: F \Rightarrow \Delta c\) is called a cone under a diagram \(F\) with nadir \(c\), or alternately is called a cocone.

Examples of Cones

Consider a diagram \(F\) indexed by the poset category \((\mathbb{Z}, \leq)\), where \(F: (n \leq m) \mapsto (f_{n,m}: F n \to F m)\). A cone over \(F\) with summit \(c\) is a family of morphisms \(\lambda_n: c \to Fn\) such that, for every \(n \leq m\), \(Ff(\lambda_j(c)) = Ff(Fj) = \lambda_k(c) = Fk\).

Limits and Colimits

A limit is the universal cone over a diagram, while a colimit is the universal cocone under a diagram.