Differences

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--- natural_transformation [2014/08/01 13:59]
nikolaj
+++ natural_transformation [2016/04/09 15:00]
nikolaj
@@ Line 2: / Line 2: @@
 ==== Collection ====
 | @#55CCEE: context     | @#55CCEE: $F,G$ in ${\bf C}\longrightarrow{\bf D}$ |
-| @#FFBB00: definiendum | @#FFBB00: $\eta$ in $\mathrm{nat}(F,G)$ |
+| @#FFBB00: definiendum | @#FFBB00: $\eta$ in $F\xrightarrow{\bullet}G$ |
 | @#AAFFAA: inclusion   | @#AAFFAA: $\eta:{\large\prod}_{(A:\mathrm{Ob}_{\bf C})}F\,A\to G\,A$ |
 | @#55EE55: postulate   | @#55EE55: @#88DDEE: $\eta\circ F(\,f)=G(\,f)\circ\eta$ |
+Here, in the postulate, I've left the components ($\eta_A,\eta_B$ etc.) implicit.
 ==== Discussion ====
-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 $A$.
+=== 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 $A$.
 === Reference ===