4.4 Adjoint functors

Adjoint functors are like matchmakers between categories, pairing up objects and morphisms in a special way. They help us see connections between different mathematical structures and give us tools to move between them.

Left and right adjoints have unique properties, preserving certain structures as they map between categories. Understanding adjoint functors is key to grasping how different mathematical worlds relate to each other.

Adjoint Functors and Adjoints

Defining Adjoint Functors

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• Adjoint 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 $\mathcal{D}$
• Adjoint functors establish a relationship between the morphisms in the two categories
• The existence of an adjunction implies that the categories $\mathcal{C}$ and $\mathcal{D}$ share similar properties and structures
• Adjoint functors play a crucial role in understanding the connections and similarities between different mathematical structures

Left and Right Adjoints

• If $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ form an adjoint pair, $F$ is called the left adjoint of $G$, and $G$ is called the right adjoint of $F$
• The left adjoint $F$ preserves colimits (coproducts, coequalizers) from $\mathcal{C}$ to $\mathcal{D}$
• The right adjoint $G$ preserves limits (products, equalizers) from $\mathcal{D}$ to $\mathcal{C}$
• The existence of a left or right adjoint for a functor provides additional properties and insights into the functor's behavior (preservation of certain categorical constructions)

Adjunction Isomorphism

• An adjunction between functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ is characterized by a natural isomorphism: $\text{Hom}_{\mathcal{D}}(F(C), D) \cong \text{Hom}_{\mathcal{C}}(C, G(D))$ for all objects $C$ in $\mathcal{C}$ and $D$ in $\mathcal{D}$
• The adjunction isomorphism relates the morphisms between objects in the categories $\mathcal{C}$ and $\mathcal{D}$
• The adjunction isomorphism is natural, meaning it commutes with the composition of morphisms in both categories
• Examples of adjoint functors include:
  • The free functor $F: \mathbf{Set} \to \mathbf{Grp}$ (from sets to groups) and the forgetful functor $G: \mathbf{Grp} \to \mathbf{Set}$
  • The product functor $\Pi: \mathbf{Set}^I \to \mathbf{Set}$ (from the category of $I$-indexed sets to sets) and the diagonal functor $\Delta: \mathbf{Set} \to \mathbf{Set}^I$

Universal Properties and Units

Universal Properties

• A universal property characterizes an object or a morphism in a category by its relationship with other objects and morphisms
• Universal properties describe the unique existence of morphisms that satisfy certain conditions
• Objects or morphisms satisfying a universal property are unique up to unique isomorphism
• Universal properties provide a way to define and characterize mathematical structures categorically
• Examples of universal properties include:
  • The product of objects in a category (characterized by the existence of unique morphisms from any other object to the product)
  • The coproduct of objects in a category (characterized by the existence of unique morphisms from the coproduct to any other object)

Unit and Counit of Adjunction

• Given an adjunction between functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$, there exist natural transformations called the unit and counit of the adjunction
• The unit of the adjunction is a natural transformation $\eta: 1_{\mathcal{C}} \to GF$, where $1_{\mathcal{C}}$ is the identity functor on $\mathcal{C}$
• The counit of the adjunction is a natural transformation $\varepsilon: FG \to 1_{\mathcal{D}}$, where $1_{\mathcal{D}}$ is the identity functor on $\mathcal{D}$
• The unit and counit satisfy the triangle identities:
  • $(\varepsilon F) \circ (F \eta) = 1_F$
  • $(G \varepsilon) \circ (\eta G) = 1_G$
• The unit and counit provide a way to relate the compositions of the adjoint functors $F$ and $G$ to the identity functors on their respective categories

Adjunctions and Universal Morphisms

• Adjunctions can be expressed in terms of universal morphisms
• The unit of an adjunction corresponds to a universal morphism from an object to its image under the right adjoint functor
• The counit of an adjunction corresponds to a universal morphism from the image of an object under the left adjoint functor to the object itself
• The universal property of the unit and counit characterizes the adjunction
• Examples of universal morphisms in adjunctions include:
  • The unit of the free-forgetful adjunction between sets and groups (the inclusion of a set into its free group)
  • The counit of the product-diagonal adjunction between sets and indexed sets (the projection from a product to its components)

Equivalence of Categories

Defining Equivalence of Categories

• An equivalence of categories is a stronger notion than an isomorphism of categories
• Two categories $\mathcal{C}$ and $\mathcal{D}$ are equivalent if there exist functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ such that:
  • There are natural isomorphisms $\alpha: GF \to 1_{\mathcal{C}}$ and $\beta: FG \to 1_{\mathcal{D}}$
  • The compositions $GF$ and $FG$ are naturally isomorphic to the identity functors on $\mathcal{C}$ and $\mathcal{D}$, respectively
• Equivalent categories have the same categorical properties and structure, but may differ in their object and morphism sets
• Equivalence of categories preserves limits, colimits, and other categorical constructions
• Examples of equivalent categories include:
  • The category of finite-dimensional vector spaces over a field $k$ and the category of $n \times n$ matrices over $k$
  • The category of sets and the category of complete atomic Boolean algebras

Adjoint Equivalence

• An adjoint equivalence is a special case of an equivalence of categories
• In an adjoint equivalence, the functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ form an adjoint pair
• The unit and counit of the adjunction are natural isomorphisms
• Adjoint equivalences provide a way to establish the equivalence of categories using the properties of adjoint functors
• Examples of adjoint equivalences include:
  • The equivalence between the category of finite-dimensional vector spaces over a field $k$ and the category of finite-dimensional $k$-algebras
  • The equivalence between the category of compact Hausdorff spaces and the category of commutative $C^*$-algebras

Applications of Equivalence

• Equivalence of categories allows the transfer of results and properties between equivalent categories
• Equivalent categories can be used interchangeably in mathematical reasoning and problem-solving
• Equivalence of categories simplifies the study of complex structures by relating them to simpler or better-understood categories
• Equivalence of categories is used in various areas of mathematics, including algebra, topology, and geometry, to establish connections and similarities between different mathematical objects and structures