Why determinant is a natural transformation?

I am reading a tutorial on Categories and it says that determinant $GL_n \longrightarrow \left( \right)^*$ is a simple example of a natural transformation, but embarrassingly I am a bit confused about it. What is this $\left( \right)^*$ means, what are our two categories here? Thanks.

Solution 1:

$\mathrm{GL}_n(-)$ is a functor from the category $\textbf{CRing}$ of commutative rings to the category $\textbf{Grp}$ of groups, mapping a commutative ring $A$ to the group $\mathrm{GL}_n(A)$. Similarly, $(-)^*$ (I prefer to write $(-)^\times$) is a functor $\textbf{CRing} \to \textbf{Grp}$ sending a ring $A$ to its group of units $A^*$. One then checks that the determinant is a natural transformation of these functors.

Solution 2:

$()^*$ is the functor that takes a ring $R$ and returns its group of units $R^*$. It is common to write functors using notation that reflects the way its values are written: since $R^*$ is the group of units of $R$, we write $()^*$ for the corresponding functor. Actually, $(-)^*$ is more common.

Other examples you see might be something like $\hom(X, -)$ which is the functor that sends an object $Y$ of your category to the set $\hom(X, Y)$, and a corresponding action on morphisms $f:Y \to Z$.

Anyways, both listed functors are from the category of commutative rings to the category of groups.