Definition:Natural Transformation/Contravariant Functors
Jump to navigation
Jump to search
Definition
Let $\mathbf C$ and $\mathbf D$ be categories.
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:
- $\xymatrix{ \map F x \ar[d]^{\eta_x} & \map F y \ar[d]^{\eta_y} \ar[l]^{\map F f} \\ \map G x & \map G y \ar[l]^{\map G f} }$
Also see
Sources
There are no source works cited for this page. Source citations are highly desirable, and mandatory for all definition pages. Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page. |