Natural isomorphism - Knowlpedia

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

A natural isomorphism $\eta:F\Rightarrow G$ is a natural transformation such that for every object $X\in\mathcal C$ , the component

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

is an isomorphism in $\mathcal D$ .

Equivalently, $\eta$ is a natural isomorphism iff there exists a natural transformation $\eta^{-1}:G\Rightarrow F$ such that $\eta^{-1}_X=(\eta_X)^{-1}$ for all $X$ (and hence $\eta^{-1}\circ \eta=\mathrm{id}_F$ , $\eta\circ \eta^{-1}=\mathrm{id}_G$ ).

Examples

Double dual on finite-dimensional vector spaces. In $\mathbf{FinVect}_k$ , the canonical maps $V\to V^{\ast\ast}$ assemble to a natural transformation $\mathrm{Id}\Rightarrow (-)^{\ast\ast}$ , and each component is an isomorphism. Hence $\mathrm{Id}\;\cong\;(-)^{\ast\ast}$ naturally on $\mathbf{FinVect}_k$ .

Swap of factors for products. In any category with binary products, there is a natural isomorphism $\sigma_{A,B}:A\times B \xrightarrow{\cong} B\times A$ characterized by $\pi_1\circ \sigma_{A,B}=\pi_2$ and $\pi_2\circ \sigma_{A,B}=\pi_1$ . In $\mathbf{Set}$ it is the function $(a,b)\mapsto(b,a)$ ; in $\mathbf{Grp}$ it is the group homomorphism $(g,h)\mapsto(h,g)$ .

Swap of summands for coproducts. Dually, in any category with binary coproducts, there is a natural isomorphism $A\sqcup B \xrightarrow{\cong} B\sqcup A$ induced by the universal property of the coproduct. In $\mathbf{Set}$ this is the obvious bijection between disjoint unions; in $\mathbf{Ab}$ it is the canonical isomorphism $A\oplus B\cong B\oplus A$ .