Cone (category Theory) - Equivalent Formulations

Equivalent Formulations

At first glance cones seem to be slightly abnormal constructions in category theory. They are maps from an object to a functor (or vice-versa). In keeping with the spirit of category theory we would like to define them as morphisms or objects in some suitable category. In fact, we can do both.

Let J be a small category and let CJ be the category of diagrams of type J in C (this is nothing more than a functor category). Define the diagonal functor Δ : C → CJ as follows: Δ(N) : J → C is the constant functor to N for all N in C.

If F is a diagram of type J in C, the following statements are equivalent:

• ψ is a cone from N to F
• ψ is a natural transformation from Δ(N) to F
• (N, ψ) is an object in the comma category (Δ ↓ F)

The dual statements are also equivalent:

• ψ is a co-cone from F to N
• ψ is a natural transformation from F to Δ(N)
• (N, ψ) is an object in the comma category (F ↓ Δ)

These statements can all be verified by a straightforward application of the definitions. Thinking of cones as natural transformations we see that they are just morphisms in CJ with source (or target) a constant functor.