🧮Topos Theory Unit 4 – Adjoint Functors

What's the Big Idea?

• Adjoint functors capture a fundamental relationship between two categories
• They consist of a pair of functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ that are "adjoint" to each other
• Adjoint functors establish a correspondence between morphisms in one category and morphisms in another category
• This correspondence is natural in the sense that it respects the composition of morphisms
• Adjoint functors provide a way to compare and relate different mathematical structures
• They allow for the transfer of properties and constructions between categories
• Adjoint functors play a central role in many areas of mathematics, including topology, algebra, and logic

Key Concepts

• Categories: Mathematical structures consisting of objects and morphisms between them
• Functors: Structure-preserving maps between categories that map objects to objects and morphisms to morphisms
• Natural transformations: A way to compare functors by providing a family of morphisms between the functors that commute with the functors' action on morphisms
• Hom-sets: The set of morphisms between two objects in a category
• Isomorphisms: Morphisms that have an inverse, indicating a strong equivalence between objects
• Universal properties: A way to characterize objects and morphisms in terms of their relationships with other objects and morphisms
• Limits and colimits: Constructions that generalize notions like products, coproducts, equalizers, and coequalizers

Formal Definition

• Given categories $\mathcal{C}$ and $\mathcal{D}$, an adjunction between $\mathcal{C}$ and $\mathcal{D}$ consists of:
  • Functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$
  • A natural isomorphism $\Phi: \text{Hom}_\mathcal{D}(F-, -) \to \text{Hom}_\mathcal{C}(-, G-)$
• The functor $F$ is called the left adjoint, and the functor $G$ is called the right adjoint
• For objects $X \in \mathcal{C}$ and $Y \in \mathcal{D}$, the isomorphism $\Phi$ provides a bijection between the hom-sets:
  • $\Phi_{X,Y}: \text{Hom}_\mathcal{D}(FX, Y) \to \text{Hom}_\mathcal{C}(X, GY)$
• The naturality of $\Phi$ means that for morphisms $f: X \to X'$ in $\mathcal{C}$ and $g: Y \to Y'$ in $\mathcal{D}$, the following diagram commutes:

Hom(FX, Y)  --Φ_{X,Y}-->  Hom(X, GY)
    |                          |
Hom(Ff, g)                 Hom(f, Gg)
    |                          |
Hom(FX', Y') --Φ_{X',Y'}--> Hom(X', GY')

Properties and Characteristics

• Adjoint functors are unique up to natural isomorphism
• If $F$ is left adjoint to $G$, then $G$ is right adjoint to $F$
• Adjoint functors preserve certain limits and colimits:
  • Left adjoints preserve colimits
  • Right adjoints preserve limits
• Adjoint functors induce a monad on the domain category and a comonad on the codomain category
• The unit and counit of an adjunction provide natural transformations between the identity functors and the compositions of the adjoint functors:
  • Unit: $\eta: 1_\mathcal{C} \to GF$
  • Counit: $\varepsilon: FG \to 1_\mathcal{D}$
• Adjoint functors satisfy the triangle identities, which relate the unit and counit:
  • $G\varepsilon \circ \eta G = 1_G$
  • $\varepsilon F \circ F\eta = 1_F$

Examples in Action

• The free-forgetful adjunction:
  • The free functor $F: \mathbf{Set} \to \mathbf{Grp}$ assigns to each set the free group generated by that set
  • The forgetful functor $G: \mathbf{Grp} \to \mathbf{Set}$ assigns to each group its underlying set of elements
• The product-exponential adjunction in a cartesian closed category:
  • The product functor $- \times A: \mathcal{C} \to \mathcal{C}$ is left adjoint to the exponential functor $(-)^A: \mathcal{C} \to \mathcal{C}$
• The direct image-inverse image adjunction in topology:
  • For a continuous map $f: X \to Y$, the direct image functor $f_*: \mathbf{Sh}(X) \to \mathbf{Sh}(Y)$ is left adjoint to the inverse image functor $f^*: \mathbf{Sh}(Y) \to \mathbf{Sh}(X)$, where $\mathbf{Sh}(-)$ denotes the category of sheaves
• The tensor-hom adjunction in a closed monoidal category:
  • The tensor functor $- \otimes A: \mathcal{C} \to \mathcal{C}$ is left adjoint to the internal hom functor $[A, -]: \mathcal{C} \to \mathcal{C}$

Connections to Other Topics

• Adjoint functors are closely related to the concept of representable functors
  • A functor $F: \mathcal{C} \to \mathbf{Set}$ is representable if it is naturally isomorphic to a hom-functor $\text{Hom}_\mathcal{C}(X, -)$ for some object $X \in \mathcal{C}$
• Adjoint functors play a crucial role in the theory of monads and comonads
  • Every adjunction induces a monad on the domain category and a comonad on the codomain category
  • Conversely, every monad and comonad arise from an adjunction
• Adjoint functors are used to define and study various types of algebras and coalgebras
  • Algebras for a monad are objects in the domain category equipped with a compatible action of the monad
  • Coalgebras for a comonad are objects in the codomain category equipped with a compatible coaction of the comonad
• Adjoint functors provide a way to relate different mathematical structures and transfer properties between them
  • They are used to establish equivalences and comparisons between categories
  • Adjoint functors can be used to define and study derived functors, which measure the failure of a functor to preserve certain limits or colimits

Common Pitfalls

• Confusing left and right adjoints
  • It's important to keep track of which functor is the left adjoint and which is the right adjoint, as they have different properties and preserve different types of limits and colimits
• Forgetting the naturality condition
  • The isomorphism between hom-sets in an adjunction must be natural, meaning it must commute with the functors' action on morphisms
• Neglecting the triangle identities
  • The unit and counit of an adjunction must satisfy the triangle identities, which ensure that they are compatible with each other and the adjoint functors
• Misunderstanding the uniqueness of adjoints
  • Adjoint functors are unique up to natural isomorphism, but there may be different choices of unit and counit that give rise to the same adjunction
• Overlooking the connection between adjoint functors and universal properties
  • Many constructions in mathematics can be characterized by universal properties, which often arise from adjoint functors
  • Recognizing the role of adjoint functors can provide insight into the nature and properties of these constructions

Why It Matters

• Adjoint functors provide a powerful framework for comparing and relating different mathematical structures
  • They allow for the transfer of properties, constructions, and intuition between categories
  • Adjoint functors can be used to establish equivalences and correspondences between seemingly disparate areas of mathematics
• Adjoint functors are a unifying concept that appears in many branches of mathematics
  • They play a central role in algebra, topology, geometry, logic, and theoretical computer science
  • Understanding adjoint functors can provide a deeper understanding of the connections and analogies between these fields
• Adjoint functors are closely related to other important categorical concepts, such as monads, comonads, and universal properties
  • Mastering adjoint functors can help in understanding and applying these concepts effectively
• Adjoint functors have practical applications in areas such as optimization, constraint satisfaction, and programming language semantics
  • They provide a theoretical foundation for techniques such as Lagrangian duality, Galois connections, and type systems
• Recognizing and leveraging adjoint functors can lead to more elegant and efficient solutions to mathematical problems
  • Adjoint functors often encapsulate the essential properties and relationships needed to solve a problem
  • By working with adjoint functors, one can take advantage of their properties and avoid unnecessary complications