Definition:Natural Transformation
Definition
Let $\mathbf C$ and $\mathbf D$ be categories.
Covariant Functors
Let $F, G : \mathbf C \to \mathbf D$ be covariant functors.
A natural transformation $\eta$ from $F$ to $G$ is a mapping on $\mathbf C$ such that:
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- $(1): \quad$ For all $x \in \mathbf C$, $\eta_x$ is a morphism from $\map F x$ to $\map G x$.
- $(2): \quad$ For all $x, y \in C$ and morphism $f: x \to y$, the following diagram commutes:
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$\quad \quad \xymatrix{ \map F x \ar[d]^{\eta_x} \ar[r]^{\map F f} & \map F y \ar[d]^{\eta_y} \\ \map G x \ar[r]^{\map G f} & \map G y}$
Contravariant Functors
Let $F, G: \mathbf C \to \mathbf D$ be contravariant functors.
A natural transformation $\eta$ from $F$ to $G$ is a mapping on $\mathbf C$ such that:
- $(1): \quad$ For all $x \in \mathbf C$, $\eta_x$ is a morphism from $\map F x$ to $\map G x$.
- $(2): \quad$ For all $x, y \in C$ and morphism $f: x \to y$, the following diagram commutes:
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$\quad \quad \xymatrix{ \map F x \ar[d]^{\eta_x} & \map F y \ar[d]^{\eta_y} \ar[l]^{\map F f} \\ \map G x & \map G y \ar[l]^{\map G f} }$
Also see
Historical Note
The concept of a natural transformation was introduced by Samuel Eilenberg and Saunders Mac Lane.
It came about during their investigations into vector spaces and their duals.
Sources
- 1964: Peter Freyd: Abelian Categories ... (previous) ... (next): Introduction