Functors and equivalence are key concepts in category theory, linking different mathematical structures. They help us understand how categories relate to each other, with essentially surjective functors "reaching" all objects and fully faithful functors preserving morphism structures.

Equivalence of categories is a powerful tool, combining essential surjectivity and full faithfulness. It allows us to view seemingly different categories as essentially the same, preserving both objects and morphisms. This concept is crucial for understanding deep connections in mathematics.

Functors and Equivalence

Definition of essentially surjective functors

• A functor $F: \mathcal{C} \to \mathcal{D}$ is essentially surjective if for every object $D$ in the codomain category $\mathcal{D}$, there exists an object $C$ in the domain category $\mathcal{C}$ such that $F(C)$ is isomorphic to $D$
  • Intuitively means every object in the codomain category is "essentially" reached by the functor, up to isomorphism
• The forgetful functor $U: \mathbf{Grp} \to \mathbf{Set}$ is essentially surjective
  • For any set $X$ in $\mathbf{Set}$, there exists a group $(X, *)$ in $\mathbf{Grp}$ such that $U((X, *)) = X$
• The functor $F: \mathbf{Ab} \to \mathbf{Grp}$ that includes abelian groups into groups is essentially surjective
  • Every group is isomorphic to an abelian group, so the functor "essentially" reaches all objects in $\mathbf{Grp}$

Definition of fully faithful functors

• A functor $F: \mathcal{C} \to \mathcal{D}$ is fully faithful if for any two objects $C_1, C_2$ in the domain category $\mathcal{C}$, the map $F_{C_1, C_2}: \text{Hom}{\mathcal{C}}(C_1, C_2) \to \text{Hom}{\mathcal{D}}(F(C_1), F(C_2))$ is bijective
  • Intuitively means the functor induces a bijection between the hom-sets of the source and target categories, preserving the morphism structure
• The inclusion functor $I: \mathbf{Ab} \to \mathbf{Grp}$ is fully faithful
  • For any two abelian groups $A, B$, the group homomorphisms between $A$ and $B$ are exactly the same as the group homomorphisms between $I(A)$ and $I(B)$
• The forgetful functor $U: \mathbf{Vect}_{\mathbb{R}} \to \mathbf{Set}$ is not fully faithful
  • There exist functions between the underlying sets of two vector spaces that are not linear transformations, so the functor does not induce a bijection on hom-sets

Equivalence of functors

• A functor $F: \mathcal{C} \to \mathcal{D}$ is an equivalence if and only if it is both fully faithful and essentially surjective
• If $F$ is an equivalence, then:
  1. By definition, $F$ has a quasi-inverse $G: \mathcal{D} \to \mathcal{C}$ such that $FG \cong 1_{\mathcal{D}}$ and $GF \cong 1_{\mathcal{C}}$
  2. Essential surjectivity follows from $FG \cong 1_{\mathcal{D}}$, as every object in $\mathcal{D}$ is isomorphic to an object in the image of $F$
  3. Full faithfulness follows from the isomorphisms $FG \cong 1_{\mathcal{D}}$ and $GF \cong 1_{\mathcal{C}}$, which induce bijections on hom-sets
• If $F$ is fully faithful and essentially surjective, then:
  1. Essential surjectivity allows choosing, for each $D \in \mathcal{D}$, an object $C_D \in \mathcal{C}$ and an isomorphism $\alpha_D: F(C_D) \to D$
  2. Define a functor $G: \mathcal{D} \to \mathcal{C}$ by $G(D) = C_D$ and $G(f) = F^{-1}(\alpha_{D'} \circ f \circ \alpha_D^{-1})$ for $f: D \to D'$
  3. The natural isomorphisms $\alpha: FG \to 1_{\mathcal{D}}$ and $\beta: GF \to 1_{\mathcal{C}}$ (induced by full faithfulness) make $G$ a quasi-inverse of $F$, proving $F$ is an equivalence

Natural isomorphisms in surjective functors

• Natural isomorphisms play a crucial role in defining essential surjectivity
  • A functor $F: \mathcal{C} \to \mathcal{D}$ is essentially surjective if for every $D \in \mathcal{D}$, there exists $C \in \mathcal{C}$ and an isomorphism $\alpha_D: F(C) \to D$
  • The collection of these isomorphisms $\alpha_D$ forms a natural isomorphism $\alpha: F \circ G \to 1_{\mathcal{D}}$, where $G: \mathcal{D} \to \mathcal{C}$ is a quasi-inverse of $F$
• The natural isomorphism $\alpha: F \circ G \to 1_{\mathcal{D}}$ encodes the idea that every object in $\mathcal{D}$ is "essentially" reached by $F$, up to isomorphism
• Naturality of $\alpha$ ensures the isomorphisms $\alpha_D$ are compatible with the morphisms in $\mathcal{C}$ and $\mathcal{D}$
  • For any morphism $f: D \to D'$ in $\mathcal{D}$, the following diagram commutes: $\begin{CD} F(G(D)) @>F(G(f))>> F(G(D')) \\ @V\alpha_DVV @VV\alpha_{D'}V \\ D @>>f> D' \end{CD}$