Natural transformation - Imported Knowlpedia Content

Let $\mathcal C,\mathcal D$ be categories, and let $F,G:\mathcal C\to\mathcal D$ be functors.

A natural transformation $\eta:F\Rightarrow G$ assigns to every object $X\in\mathcal C$ a morphism

\eta_X:F(X)\to G(X)

in $\mathcal D$ , such that for every morphism $f:X\to Y$ in $\mathcal C$ , the naturality condition

G(f)\circ \eta_X \;=\; \eta_Y\circ F(f)

holds (composition in $\mathcal D$ ; see composition).

Equivalent characterizations

Equivalently, for each $f:X\to Y$ the square commutes:

Remarks

The components $\eta_X$ are called the components of $\eta$ .

Examples

Singleton map $X\to\mathcal P(X)$ (Set). Let $\mathbf{Set}$ be the category of sets. Define the covariant “power set” functor $P:\mathbf{Set}\to\mathbf{Set}$ by $P(X)=\mathcal P(X)$ and, for a function $f:X\to Y$ , let $P(f):\mathcal P(X)\to\mathcal P(Y)$ be the image map $S\mapsto f(S)$ . Then the family $\eta_X:X\to\mathcal P(X)$ , $x\mapsto\{x\}$ , defines a natural transformation $\eta:\mathrm{Id}_{\mathbf{Set}}\Rightarrow P,$ since $P(f)(\{x\})=\{f(x)\}$ .

Postcomposition induces a natural transformation on representables. In any $\mathcal C$ , fix a morphism $h:A\to B$ . Consider the contravariant hom-functors $\mathrm{Hom}_{\mathcal C}(-,A)$ and $\mathrm{Hom}_{\mathcal C}(-,B)$ (see contravariant functor). Define, for each $X$ , $\eta_X:\mathrm{Hom}_{\mathcal C}(X,A)\to \mathrm{Hom}_{\mathcal C}(X,B),\quad f\mapsto h\circ f.$ Naturality follows from associativity of composition.

The evaluation map into the double dual (Vect $_k$ ). In the category of $k$ -vector spaces, there is a natural transformation $\mathrm{ev}:\mathrm{Id}\Rightarrow (-)^{\ast\ast}$ whose component at $V$ is the canonical linear map $V\to V^{\ast\ast}$ , $v\mapsto(\varphi\mapsto\varphi(v))$ . For finite-dimensional $V$ this component is an isomorphism, so $\mathrm{ev}$ becomes a natural isomorphism on the full subcategory of finite-dimensional spaces.