natural_transformation

Natural transformation

Collection

context	$F,G$ in ${\bf C}\longrightarrow{\bf D}$
definiendum	$\eta$ in $F\xrightarrow{\bullet}G$
inclusion	$\eta:{\large\prod}_{(A:\mathrm{Ob}_{\bf C})}F\,A\to G\,A$
postulate	$\eta\circ F(\,f)=G(\,f)\circ\eta$

Here, in the postulate, I've left the components ( $\eta_A,\eta_B$ etc.) implicit.

Discussion

Idea

Natural transformation form a collection of arrows within a single category which are compatible with the (structure preserving) functors.

Elaboration

If one thinks about it for a minute, the data provided with a natural transformation can in fact be reformulated as just another functor, namely in ${\bf C}\times(\bullet\to\bullet)\longrightarrow{\bf D}$ . This mirrors a homotopy.

Notation

For any $A:\mathrm{Ob}_{\bf C}$ , we write $\eta_A$ for the map $F\,A\to G\,A$ . This is called the component of the natural transformation $\eta$ at $A$ .