5.3 Natural isomorphisms and equivalences

Natural isomorphisms are special natural transformations where each component is an isomorphism. They're crucial for understanding when functors are essentially the same, differing only in specific mappings. This concept helps us compare and relate different mathematical structures.

Equivalence of categories takes this idea further, showing when two categories have the same structure despite different objects and morphisms. This powerful tool allows us to transfer properties and results between seemingly different mathematical contexts, revealing deeper connections.

Natural Isomorphisms and Equivalences

Natural isomorphisms and inverses

• A natural transformation $\alpha: F \to G$ is a natural isomorphism when each component $\alpha_A: F(A) \to G(A)$ is an isomorphism in the target category for every object $A$ in the source category
  • The inverse of a natural isomorphism $\alpha$ is a natural transformation $\beta: G \to F$ where each component $\beta_A: G(A) \to F(A)$ is the inverse of $\alpha_A$ in the target category for every object $A$
    • The compositions $\alpha \circ \beta$ and $\beta \circ \alpha$ equal the identity natural transformations $1_F$ and $1_G$, respectively ($\alpha \circ \beta = 1_F$ and $\beta \circ \alpha = 1_G$)
  • Example: The double dual isomorphism $V \to (V^*)^*$ for finite-dimensional vector spaces over a field $k$ is a natural isomorphism between the identity functor and the double dual functor

Relationship to isomorphic functors

• Two functors $F, G: \mathcal{C} \to \mathcal{D}$ are isomorphic when there exists a natural isomorphism $\alpha: F \to G$ between them
  • Isomorphic functors are essentially the same, only differing in their specific choice of object and morphism mappings
• For isomorphic functors $F$ and $G$, the induced functions $F_{A,B}: \text{Hom}_\mathcal{C}(A, B) \to \text{Hom}_\mathcal{D}(F(A), F(B))$ and $G_{A,B}: \text{Hom}_\mathcal{C}(A, B) \to \text{Hom}_\mathcal{D}(G(A), G(B))$ are bijections for any objects $A, B$ in $\mathcal{C}$
  • Example: The identity functor $1_{\mathbf{Vect}_k}: \mathbf{Vect}_k \to \mathbf{Vect}_k$ and the dual space functor $(-)^*: \mathbf{Vect}_k^{\text{op}} \to \mathbf{Vect}_k$ are isomorphic functors between the category of finite-dimensional vector spaces over a field $k$ and its opposite category

Equivalence of categories

• Two categories $\mathcal{C}$ and $\mathcal{D}$ are equivalent when there exist functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ such that:
  1. There is a natural isomorphism $\alpha: 1_\mathcal{C} \to G \circ F$ between the identity functor $1_\mathcal{C}$ on $\mathcal{C}$ and the composition $G \circ F$
  2. There is a natural isomorphism $\beta: 1_\mathcal{D} \to F \circ G$ between the identity functor $1_\mathcal{D}$ on $\mathcal{D}$ and the composition $F \circ G$
• Equivalent categories have the same structure and properties but may have different objects and morphisms
  • Example: The category of finite sets ($\mathbf{FinSet}$) and the category of finite-dimensional vector spaces over the field with two elements $\mathbb{F}_2$ ($\mathbf{Vect}_{\mathbb{F}_2}$) are equivalent categories

Examples of natural isomorphisms

• The double dual isomorphism $V \to (V^*)^*$ for finite-dimensional vector spaces over a field $k$ is a natural isomorphism between the identity functor and the double dual functor on $\mathbf{Vect}_k$
• The exponential and logarithm functions form a natural isomorphism between the functors $(-)^+: \mathbf{Top} \to \mathbf{Top}$ (adding a disjoint basepoint) and $\Sigma(-): \mathbf{Top} \to \mathbf{Top}$ (reduced suspension) on the category of topological spaces
• The functors $F: \mathbf{FinSet} \to \mathbf{Vect}_{\mathbb{F}_2}$ (sending each set to its $\mathbb{F}_2$-vector space of functions) and $G: \mathbf{Vect}_{\mathbb{F}_2} \to \mathbf{FinSet}$ (sending each vector space to its underlying set) establish an equivalence between the categories $\mathbf{FinSet}$ and $\mathbf{Vect}_{\mathbb{F}_2}$ via natural isomorphisms constructed using the bijection between sets and their characteristic functions