natural transformation

Idea

• just as a functor is a morphism between categories, a natural transformation is a 2-morphism between functors.

Definition

• Given category \(C\) and \(D\) and functors \(F, G: C \rightarrow D\)
• a natural transformation \(\alpha: F \Rightarrow G\) between them, denoted is an assignment to each object \(x\) in \(C\) a morphism \(\alpha_x: F(x) \rightarrow G(x)\) in \(D\), called the component of \(\alpha\) at \(x\),
• such that the following diagram commutes
• for \(x \xrightarrow{f} y\) in \(C\)
  • items on top row are given by functor \(F\). They are all in \(D\)
  • bottom row given by functor \(G\).

Composition of natural transformation

• Given
  • three functors \(F, G, H: C \rightarrow D\),
  • two natural transformations \(\alpha: F \Rightarrow G\) and \(\beta: G \Rightarrow H\)
• composition, \(F \Rightarrow H\) , is given obviously.

Alternative definition: in terms of morphism-wise components

• assign every morphism in \(C\) a morphism in \(D\) such that ...
• ultimately equivalent

Quiver links:

• 1