Cone (category theory)

Let F : J → C be a diagram in C. Formally, a diagram is nothing more than a functor from J to C. The change in terminology reflects the fact that we think of F as indexing a family of objects and morphisms in C. The category J is thought of as an "index category".

Thus, for example, when J is a discrete category, it corresponds most closely to the idea of an indexed family in set theory.

Thinking of cones as natural transformations we see that they are just morphisms in CJ with source (or target) a constant functor.

This equivalence is rooted in the observation that a natural map between constant functors Δ(N), Δ(M) corresponds to a morphism between N and M. In this sense, the diagonal functor acts trivially on arrows.

In similar vein, writing down the definition of a natural map from a constant functor Δ(N) to F yields the same diagram as the above.