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 JJ and CC and constant functor Δ(c):JC\Delta(c):J\to C and a diagram F:JCF:J\to C, a natural transformation α:Δ(c)F\alpha:\Delta(c) \Rightarrow F

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

assigns to every object jj in JJ a morphism α x:cF(j)\alpha_x:c \to F(j) in DD (called the component of α\alpha at jj) such that for any morphism f:jkf:j \to k in JJ, the following diagram commutes in DD:

c Id c c α x α y F(j) F(f) F(j). \array{ c & \stackrel{Id_c}{\to} & c \\ \alpha_x\downarrow && \downarrow \alpha_y \\ F(j) & \stackrel{F(f)}{\to} & F(j) } \,.


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

If cc is an object of CC, a cone from cc to FF is a natural transformation

T:Δ(c)FT: \Delta(c) \to F

where Δ(c):JC\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:cF(j),T_j: c \to F(j),

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

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


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