Eric Forgy Cone

Warning: These pages are just my notes trying to unwrap the definition of a cone in terms of natural transformations to components. Feedback welcome!

Given categories $J$ and $C$ and constant functor $\Delta(c):J\to C$ and a diagram $F:J\to C$, a natural transformation $\alpha:\Delta(c) \Rightarrow F$

assigns to every object $j$ in $J$ a morphism $\alpha_x:c \to F(j)$ in $D$ (called the component of $\alpha$ at $j$) such that for any morphism $f:j \to k$ in $J$, the following diagram commutes in $D$:

Definition

Let $F: J \to C$ be a diagram in a category $C$.

If $c$ is an object of $C$, a cone from $c$ to $F$ is a natural transformation

where $\Delta(c):J\to C$ denotes the constant functor.

In other words, a cone consists of morphisms (called the components of the cone)

one for each object $j$ of $J$, which are compatible with all the morphisms $F(f): F(j) \to F(k)$ of the diagram, in the sense that each diagram

commutes.