Functors are the backbone of category theory, allowing us to compare and connect different mathematical structures. In this section, we dive into three key properties of functors: faithfulness, fullness, and essential surjectivity.

These properties are crucial for understanding equivalence of categories, a fundamental concept that helps us identify when two seemingly different mathematical structures are essentially the same. We'll explore how these properties combine to define category equivalence.

Functors and Equivalence of Categories

Definition of faithful functors

• A functor $F: C \to D$ is faithful if it is injective on the hom-sets between any two objects
  • If for any two objects $X, Y$ in $C$ and any two parallel morphisms $f, g: X \to Y$ in $C$, $F(f) = F(g)$ implies $f = g$
• The forgetful functor $U: Grp \to Set$ is faithful
  • Maps group homomorphisms to functions
  • If two group homomorphisms are mapped to the same function, they must be the same homomorphism
• The inclusion functor $I: Ab \to Grp$ from the category of abelian groups to the category of groups is faithful
  • Embeds $Ab$ as a subcategory of $Grp$

Definition of full functors

• A functor $F: C \to D$ is full if it is surjective on the hom-sets between any two objects
  • For any two objects $X, Y$ in $C$ and any morphism $h: F(X) \to F(Y)$ in $D$, there exists a morphism $f: X \to Y$ in $C$ such that $F(f) = h$
• The functor $F: Set \to Set$ that maps each set to its power set and each function to its induced function between power sets is full
• The inclusion functor $I: Ab \to Grp$ is not full
  • There are group homomorphisms between abelian groups that are not homomorphisms in the category $Ab$ (the inversion map $x \mapsto x^{-1}$ in a non-abelian group)

Definition of essentially surjective functors

• A functor $F: C \to D$ is essentially surjective if every object in the codomain category is isomorphic to an object in the image of the functor
  • For every object $Y$ in $D$, there exists an object $X$ in $C$ such that $F(X)$ is isomorphic to $Y$
• The functor $F: FinSet \to FinSet$ that maps each finite set to its double (the disjoint union of the set with itself) and each function to its induced function between the doubled sets is essentially surjective
• The inclusion functor $I: Ab \to Grp$ is not essentially surjective
  • There are groups (non-abelian groups) that are not isomorphic to any abelian group

Equivalence of categories proof

• An equivalence of categories is a functor $F: C \to D$ for which there exists a functor $G: D \to C$ such that $G \circ F \cong 1_C$ and $F \circ G \cong 1_D$
  • $1_C$ and $1_D$ are the identity functors on $C$ and $D$, respectively
• Proof:
  1. ($\Rightarrow$) If $F$ is an equivalence of categories, then it is faithful, full, and essentially surjective:
     • Faithful: If $F(f) = F(g)$, then $G(F(f)) = G(F(g))$. Since $G \circ F \cong 1_C$, we have $f = g$
     • Full: For any $h: F(X) \to F(Y)$ in $D$, there exists a morphism $k: X \to G(F(Y))$ in $C$ such that $F(k) = h \circ \varphi_Y$, where $\varphi: F \circ G \Rightarrow 1_D$ is the natural isomorphism. Then, $f = G(\varphi_Y) \circ k$ satisfies $F(f) = h$
     • Essentially surjective: For any $Y$ in $D$, $F(G(Y))$ is isomorphic to $Y$ via the natural isomorphism $\varphi: F \circ G \Rightarrow 1_D$
  2. ($\Leftarrow$) If $F$ is faithful, full, and essentially surjective, then it is an equivalence of categories:
     • Define $G: D \to C$ as follows:
       • For each $Y$ in $D$, choose an object $G(Y)$ in $C$ such that $F(G(Y))$ is isomorphic to $Y$ (which exists by essential surjectivity)
       • For each morphism $h: Y \to Y'$ in $D$, define $G(h)$ as the unique morphism $G(Y) \to G(Y')$ such that $F(G(h)) = \varphi_{Y'} \circ h \circ \varphi_Y^{-1}$, where $\varphi_Y: F(G(Y)) \to Y$ and $\varphi_{Y'}: F(G(Y')) \to Y'$ are the chosen isomorphisms (which exists by fullness and faithfulness)
     • The natural isomorphisms $\varphi: F \circ G \Rightarrow 1_D$ and $\psi: 1_C \Rightarrow G \circ F$ can be defined using the chosen isomorphisms and the fullness and faithfulness of $F$, making $F$ and $G$ quasi-inverses and thus an equivalence of categories