natural transformation

Definition

A natural transformation $\eta$ of functors $F,G:\mathcal{C}\to \mathcal{D}$ (denoted $\eta :F\Rightarrow G$) is the data of morphisms

$${\eta }_X:F(X)\to G(X)$$

in $\mathcal{D}$ such that for each morphism $f:X\to Y$ in $\mathcal{C}$, the diagram

commutes. The morphisms ${\eta }_X$, $X\in \mathcal{C}$ are called the components of $\eta$.

Natural isomorphism

A natural transformation $\eta :F\Rightarrow G$ is called a natural isomorphism if each component ${\eta }_X$ for $X\in \mathcal{C}$ is an isomorphism.

Example

Note that for the categories $\text{CRing}$ of commutative rings and $\text{Grp}$ of groups, then we have the following functors:

$$\begin{array}{rl}{\text{GL}}_n:\text{CRing}& ⟶\text{Grp}\\ R& ⟼{\text{GL}}_n(R)\\ R\stackrel{f}{\to }S& ⟼{\text{GL}}_n(R)\stackrel{{\text{GL}}_n(f)}{\to }{\text{GL}}_n(S)\end{array}$$

where ${\text{GL}}_n(f)$ is applying f to each matrix entry.

$$\begin{array}{rl}\ast :\text{CRing}& ⟶\text{Grp}\\ R& ⟼R^{\ast }\\ R\stackrel{f}{\to }S& ⟼R^{\ast }\stackrel{f^{\ast }}{\to }{S}^{\ast }\end{array}$$

where $f^{\ast }:R^{\ast }\to S^{\ast }$ is the restriction to the group of units of $R$ and $S$.

Thus, the determinant ${\det }_R:{\text{GL}}_n(R)\to R^{\ast }$ is a natural transformation: For any morphism $f:R\to S$ we have

since the determinant uses the same formula independent of the ring we have

$$f^{\ast }\circ {\det }_R={\det }_S\circ G{L}_n(f)$$