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

$$\begin{array}{c}\\ & \nearrow \searrow \Delta (c)\\ J& \Downarrow \alpha & C\\ & \searrow \nearrow F\end{array}$$\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 $D$:

(1)$$\begin{array}{ccc}c& \stackrel{{\mathrm{Id}}_c}{\to }& c\\ {\alpha }_x\downarrow & & \downarrow {\alpha }_y\\ F(j)& \stackrel{F(f)}{\to }& F(j)\end{array}.$$\array{ c & \stackrel{Id_c}{\to} & c \\ \alpha_x\downarrow && \downarrow \alpha_y \\ F(j) & \stackrel{F(f)}{\to} & F(j) } \,.

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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

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

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

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

commutes.