**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** $\alpha:\Delta(c) \Rightarrow F$

$\array{
\\
& \nearrow \searrow\mathrlap{\scriptsize{\Delta(c)}}
\\
J
&\Downarrow\mathrlap{\scriptsize{\alpha}}&
C
\\
& \searrow \nearrow\mathrlap{\scriptsize{F}}
}$

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 $D$:

$\array{
c
&
\stackrel{Id_c}{\to}
&
c
\\
\alpha_x\downarrow
&&
\downarrow \alpha_y
\\ F(j)
&
\stackrel{F(f)}{\to} & F(j)
}
\,.$

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

$T: \Delta(c) \to F$

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

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

$T_j: c \to F(j),$

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

$\array{
{}&{}&c&{}&{} \\
{}& \mathllap{\scriptsize{T_j}}\swarrow &{}& \searrow\mathrlap{\scriptsize{T_k}} &{} \\
F(j) &{}&\stackrel{F(f)}{\longrightarrow} &{}& F(k) \\
}$

commutes.

Created on November 4, 2009 at 06:15:33
by
Eric Forgy