🔢Category Theory Unit 9 – Adjunctions

What Are Adjunctions?

• Adjunctions are a fundamental concept in category theory that describe a relationship between two functors
• Consist of a pair of functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ between categories $\mathcal{C}$ and $\mathcal{D}$
• Functors $F$ and $G$ are called adjoint functors or an adjoint pair
• Adjunctions capture the idea of an optimal solution to a universal mapping problem
• Establish a correspondence between morphisms in $\mathcal{C}$ and morphisms in $\mathcal{D}$
  • Specifically, there is a bijection between the sets $\text{Hom}_{\mathcal{D}}(F(C), D)$ and $\text{Hom}_{\mathcal{C}}(C, G(D))$ for objects $C$ in $\mathcal{C}$ and $D$ in $\mathcal{D}$
• Adjunctions generalize the notion of Galois connections between partially ordered sets
• Play a central role in many constructions and theorems in category theory

Key Components of Adjunctions

• Two categories $\mathcal{C}$ and $\mathcal{D}$ involved in the adjunction
• A pair of functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ between the categories
  • $F$ is called the left adjoint functor and $G$ is called the right adjoint functor
• Natural transformations $\eta: 1_{\mathcal{C}} \to GF$ (unit) and $\varepsilon: FG \to 1_{\mathcal{D}}$ (counit) that satisfy certain conditions
• Bijection between the sets of morphisms $\text{Hom}_{\mathcal{D}}(F(C), D)$ and $\text{Hom}_{\mathcal{C}}(C, G(D))$
  • This bijection is natural in both $C$ and $D$
• Adjunction isomorphism: $\text{Hom}_{\mathcal{D}}(F(C), D) \cong \text{Hom}_{\mathcal{C}}(C, G(D))$
• Triangle identities that the unit and counit must satisfy:
  • $G\varepsilon \circ \eta G = 1_G$ and $\varepsilon F \circ F\eta = 1_F$

Types of Adjunctions

• Free/forgetful adjunctions: Relate algebraic structures and their underlying sets
  • Free functor adds the "least" structure to an object, forgetful functor forgets the structure
• Limit/colimit adjunctions: Relate limits and colimits in a category
  • Diagonal functor $\Delta: \mathcal{C} \to \mathcal{C}^J$ is left adjoint to the limit functor $\text{lim}: \mathcal{C}^J \to \mathcal{C}$
• Adjunctions involving exponential objects: Relate products and exponentials in a category
  • $(-) \times A: \mathcal{C} \to \mathcal{C}$ is left adjoint to $(-)^A: \mathcal{C} \to \mathcal{C}$ for an object $A$ in a Cartesian closed category $\mathcal{C}$
• Adjunctions between functors and representable functors: Relate functors and their representing objects
  • Yoneda lemma: $\text{Hom}(-, A): \mathcal{C}^{\text{op}} \to \mathbf{Set}$ is left adjoint to $\text{Hom}(A, -): \mathcal{C} \to \mathbf{Set}$ for an object $A$ in a locally small category $\mathcal{C}$
• Adjunctions in topology: Relate continuous functions and open/closed sets
  • Functor sending a space to its lattice of open sets is right adjoint to the functor sending a lattice to its space of points

Properties of Adjoint Functors

• Uniqueness: Adjoint functors are unique up to natural isomorphism
  • If $(F, G, \eta, \varepsilon)$ and $(F', G', \eta', \varepsilon')$ are adjunctions, then $F \cong F'$ and $G \cong G'$
• Preservation of limits and colimits:
  • Left adjoints preserve colimits, right adjoints preserve limits
• Composition: Adjunctions can be composed
  • If $F \dashv G$ and $F' \dashv G'$, then $F'F \dashv GG'$
• Adjunctions induce monads:
  • Every adjunction $(F, G, \eta, \varepsilon)$ gives rise to a monad $(T, \eta, \mu)$ where $T = GF$ and $\mu = G\varepsilon F$
• Adjunctions and equivalences of categories:
  • An adjunction is an equivalence of categories if and only if the unit and counit are natural isomorphisms
• Adjunctions and universal properties:
  • Adjoint functors can be characterized by universal properties
  • Left adjoints are initial objects in certain comma categories, right adjoints are terminal objects

Examples of Adjunctions in Action

• Free group/forgetful functor adjunction:
  • Free functor $F: \mathbf{Set} \to \mathbf{Grp}$ sends a set to the free group generated by that set
  • Forgetful functor $U: \mathbf{Grp} \to \mathbf{Set}$ sends a group to its underlying set
• Product/exponential adjunction in a Cartesian closed category:
  • $(-) \times A: \mathcal{C} \to \mathcal{C}$ is left adjoint to $(-)^A: \mathcal{C} \to \mathcal{C}$ for an object $A$ in a Cartesian closed category $\mathcal{C}$
• Discrete/indiscrete topology adjunction:
  • Functor sending a set to the discrete topological space is left adjoint to the functor sending a topological space to its underlying set
• Sheafification/forgetful functor adjunction:
  • Sheafification functor $\mathbf{PSh}(X) \to \mathbf{Sh}(X)$ is left adjoint to the forgetful functor $\mathbf{Sh}(X) \to \mathbf{PSh}(X)$ for a topological space $X$
• Tensor/hom adjunction in a closed monoidal category:
  • Tensor product functor $(-) \otimes A: \mathcal{C} \to \mathcal{C}$ is left adjoint to the internal hom functor $[A, -]: \mathcal{C} \to \mathcal{C}$ for an object $A$ in a closed monoidal category $\mathcal{C}$

Applications in Other Math Fields

• Algebraic topology: Adjunctions between topological spaces and algebraic structures
  • Singular homology and cohomology functors form an adjoint pair between the category of topological spaces and the category of chain complexes
• Homological algebra: Adjunctions in the study of chain complexes and their homology
  • Tensor product and Hom functors form an adjoint pair in the category of modules over a ring
• Representation theory: Adjunctions in the study of group representations
  • Induction and restriction functors form an adjoint pair between the categories of representations of a group and its subgroup
• Algebraic geometry: Adjunctions in the study of schemes and sheaves
  • Global section functor is right adjoint to the constant sheaf functor in the category of sheaves on a scheme
• Logic and type theory: Adjunctions in the study of logical systems and their models
  • Quantifiers $\forall$ and $\exists$ form an adjoint pair between the categories of sets and propositions in a topos

Common Mistakes and Misconceptions

• Confusing left and right adjoints: Pay attention to the direction of the functors and the order of composition
• Forgetting the naturality conditions: The bijection between hom-sets must be natural in both variables
• Misunderstanding the role of the unit and counit: They are not inverses but satisfy the triangle identities
• Assuming all functors have adjoints: Not every functor has an adjoint, and the existence of an adjoint is a special property
• Misapplying preservation properties: Left adjoints preserve colimits, right adjoints preserve limits, but not vice versa in general
• Confusing adjunctions with equivalences: An adjunction is an equivalence if and only if the unit and counit are natural isomorphisms
• Overlooking the connection to universal properties: Adjoint functors can be characterized by universal properties, which can simplify proofs and constructions

Practice Problems and Solutions

1. Prove that the free group functor $F: \mathbf{Set} \to \mathbf{Grp}$ is left adjoint to the forgetful functor $U: \mathbf{Grp} \to \mathbf{Set}$.
   • Solution: Construct the unit $\eta: 1_{\mathbf{Set}} \to UF$ and counit $\varepsilon: FU \to 1_{\mathbf{Grp}}$ and show that they satisfy the triangle identities.
2. Show that the product functor $(-) \times A: \mathcal{C} \to \mathcal{C}$ is left adjoint to the exponential functor $(-)^A: \mathcal{C} \to \mathcal{C}$ for an object $A$ in a Cartesian closed category $\mathcal{C}$.
   • Solution: Use the adjunction isomorphism $\text{Hom}(X \times A, Y) \cong \text{Hom}(X, Y^A)$ and show that it is natural in both $X$ and $Y$.
3. Prove that the sheafification functor $\mathbf{PSh}(X) \to \mathbf{Sh}(X)$ is left adjoint to the forgetful functor $\mathbf{Sh}(X) \to \mathbf{PSh}(X)$ for a topological space $X$.
   • Solution: Construct the unit and counit using the universal property of sheafification and show that they satisfy the triangle identities.
4. Show that the tensor product functor $(-) \otimes A: \mathcal{C} \to \mathcal{C}$ is left adjoint to the internal hom functor $[A, -]: \mathcal{C} \to \mathcal{C}$ for an object $A$ in a closed monoidal category $\mathcal{C}$.
   • Solution: Use the adjunction isomorphism $\text{Hom}(X \otimes A, Y) \cong \text{Hom}(X, [A, Y])$ and show that it is natural in both $X$ and $Y$.
5. Prove that the singular homology and cohomology functors form an adjoint pair between the category of topological spaces and the category of chain complexes.
   • Solution: Construct the unit and counit using the natural transformations between the singular chain complex and the singular cochain complex and show that they satisfy the triangle identities.