Examples of natural isomorphisms

Question

I am a beginning Category Theory student, and a intermediate Algebra student. Could someone provide me with some examples of natural isomorphisms (in Category Theory) besides the natural isomorphism between the identity functor on Grp and the opposite functor on Grp, and the natural isomorphism between the identity functor on Vect$^{\text{finite}}_{\mathbb{F}}$ and the double dual map on Vect$^{\text{finite}}_{\mathbb{F}}$?

Answer 1

Let $C$ denote the category whose only object is $\Bbb R^n$ (for a fixed $n \in \Bbb N$) and whose morphisms are linear transformations.

Let $D$ denote the category whose only object is the set $V = \{[x_1,\dots,x_n]^T:x_i \in \Bbb R\}$ and whose morphisms are matrices (composition is given by matrix multiplication).

Given bases $A,B$ of $\Bbb R^n$ we define the functors $F,G:C \to D$ by $$ F(f) = [f]_{A \to A}\\ G(f) = [f]_{B \to B} $$ That is, our functors take a linear transformation and yield its matrix representation with respect to some basis.

We can define a natural transformation $\eta:F \to G$ by $$ \eta_V = [I]_{A \to B} $$ That is, the change-of-basis matrix with respect to $A$ and $B$. Since the matrix $\eta_V$ is invertible, $\eta$ is of course a natural isomorphism.

Answer 2

• If $G$ is a group, considered as a one-element category, then the natural isomorphisms from the identity functor to itself are exactly the automorphisms of $G$.

• If $X\to Y$ is any isomorphism in a category, then the induced natural transformation of functors $\operatorname{Hom}(-,X) \to \operatorname{Hom}(-,Y)$ is a natural isomorphism.

• You may have seen the hom-tensor adjunction $\operatorname{Hom}(Y\otimes X, Z) \cong \operatorname{Hom}(Y,\operatorname{Hom}(X,Z))$, for $R$-modules $X,Y,Z$. If we fix $X$, then the functors $\operatorname{Hom}(-\otimes X, -)$ and $\operatorname{Hom}(-,\operatorname{Hom}(X,-))$ are isomorphic functors from $C^2$ to $\operatorname{Set}$, where $C$ is the category of $R$-modules.

• That last example works for any pair of adjoint functors. You probably haven't gotten to those, but, as another example, if I take a set $S$ and a vector space $V$, then the set maps $S\to V$ are in bijection with the vector space maps $F(S) \to V$, where $F(S)$ is the free vector space with basis $S$. Then we have two functors from the category of pairs $(S,V)$ to the category of sets, namely $\operatorname{Hom}_{set}(S,V)$ and $\operatorname{Hom}_{vect}(F(S),V)$. These functors are naturally isomorphic.