Glossary

4.1 Definition and properties of adjoint functors

3 min read•july 25, 2024

Adjoint functors are a powerful tool in category theory, allowing bidirectional transformations between categories. They consist of a F and a G, connected by a natural bijection between morphism sets.

The ensures that adjoint functors are essentially unique up to natural . This property, along with the adjoint functor theorem and , makes adjoints crucial for understanding categorical structures and constructing new functors.

Adjoint Functors: Definition and Fundamental Properties

Adjoint functors between categories

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  • Adjoint functors form a powerful pair of functors between two categories enabling bidirectional transformations
    • Left adjoint F:C→DF: C \to D maps objects and morphisms from category C to D
    • Right adjoint G:D→CG: D \to C maps objects and morphisms from category D to C
  • Natural bijection establishes a correspondence between morphisms in both categories
    • HomD(F(A),B)≅HomC(A,G(B))\text{Hom}_D(F(A), B) \cong \text{Hom}_C(A, G(B)) for all objects AA in CC and BB in DD
    • Enables translation of morphisms between categories (homomorphisms, continuous functions)
  • Unit and counit serve as universal arrows in the adjunction
    • Unit η:1C→GF\eta: 1_C \to GF compares identity functor on C with composition GF
    • Counit ϵ:FG→1D\epsilon: FG \to 1_D compares composition FG with identity functor on D
  • ensure coherence of the adjunction
    • Gϵ∘ηG=1GG\epsilon \circ \eta G = 1_G verifies compatibility of unit and counit with G
    • ϵF∘Fη=1F\epsilon F \circ F\eta = 1_F verifies compatibility of unit and counit with F

Uniqueness of adjoint functors

  • Uniqueness theorem guarantees essential uniqueness of adjoint functors
    • If F⊣GF \dashv G and F⊣G′F \dashv G', then G≅G′G \cong G' natural isomorphism between right adjoints
    • If F⊣GF \dashv G and F′⊣GF' \dashv G, then F≅F′F \cong F' natural isomorphism between left adjoints
  • Proof strategy involves constructing natural isomorphisms
    • Utilize universal property of adjunctions to define component morphisms
    • Verify naturality and isomorphism conditions
  • plays crucial role in establishing uniqueness
    • Relates natural transformations to representable functors
    • Allows reduction of functor equality to object-wise equality
  • Natural transformations connect functors in the uniqueness proof
    • Between functors GG and G′G' for right adjoint uniqueness
    • Between functors FF and F′F' for left adjoint uniqueness

Applications and Implications of Adjoint Functors

Adjoint functor theorem

  • General adjoint functor theorem provides conditions for existence of left adjoints
    • Solution set condition ensures "smallness" of potential candidates
    • Applicable to wide range of categories (abelian groups, topological spaces)
  • Special adjoint functor theorem simplifies conditions for locally small, complete categories
    • Preservation of small limits replaces solution set condition
    • Useful in algebraic categories (rings, modules)
  • applies to locally presentable categories
    • Characterizes existence of left adjoints through accessibility conditions
    • Relevant in categorical logic and
  • Implications extend to various areas of mathematics
    • Constructing new functors by composing known adjoints
    • Characterizing categorical properties (limits, colimits, exponentials)

Limit preservation by adjoints

  • Right adjoints preserve limits allowing transfer of structures
    • Proof utilizes natural isomorphism of hom-sets
    • Applies to products, pullbacks, equalizers
  • Left adjoints preserve colimits enabling preservation of structures
    • Dual statement to right adjoints
    • Applies to coproducts, pushouts, coequalizers
  • Consequences impact various mathematical constructions
    • Forgetful functors often have left adjoints (free constructions)
    • Free constructions as left adjoints preserve colimits
  • Examples illustrate practical applications
    • Free group functor preserves coproducts (disjoint unions)
    • from groups to sets preserves products (Cartesian products)
  • Relationship to representable functors connects to broader category theory
    • Adjoint functors naturally isomorphic to representable functors
    • Enables application of Yoneda lemma and related results

Key Terms to Review (21)

Adjunction Theorem: The adjunction theorem describes a fundamental relationship between two functors, establishing a correspondence that captures how one functor can be seen as providing a kind of 'inverse' operation to another. This relationship is pivotal for understanding the nature of adjoint functors and is closely tied to the concepts of units and counits, which serve as natural transformations bridging the two functors in the adjoint pair. The theorem not only highlights the connection between different categories but also underscores the significance of these transformations in preserving structure within mathematical frameworks.
Colimit: A colimit is a universal construction in category theory that generalizes the concept of taking a limit of a diagram of objects and morphisms, allowing for the 'gluing' together of objects in a category. It serves as a way to define the 'largest' object that can be mapped into all objects in a given diagram, capturing the idea of combining various structures in a coherent way.
Colimit preservation: Colimit preservation refers to a property of a functor that ensures it preserves colimits from the category it maps from to the category it maps to. When a functor is said to preserve colimits, it means that if you have a diagram in the source category that has a colimit, then its image under the functor will also have a colimit in the target category that is isomorphic to the image of the original colimit.
Counit of an adjunction: The counit of an adjunction is a natural transformation that provides a way to 'coherently' transition from the functor that is the right adjoint back to the functor that is the left adjoint. In the context of an adjunction between two categories, it expresses how objects in the category related to the left adjoint can be mapped to objects in the category related to the right adjoint, capturing essential relationships between these categories.
Daniel Kan: Daniel Kan is a significant figure in category theory, best known for his contributions to the understanding of adjoint functors and their properties. His work emphasized the importance of these functors in providing a framework for relating different mathematical structures and establishing connections between various categories. Kan's insights have laid the groundwork for further developments in topos theory, allowing mathematicians to explore more complex relationships within categorical contexts.
Forgetful Functor: A forgetful functor is a type of functor that 'forgets' some structure or properties of the objects and morphisms it maps between categories, essentially providing a way to relate different categories while losing some information. It often connects categories that have a more complex structure to simpler ones, making it easier to work with and understand the relationships between various mathematical constructs.
Free Functor: A free functor is a type of functor that provides a way to generate structures in a category from a simpler or more basic one, without imposing any additional relations. It can be thought of as a way to create new objects and morphisms by freely generating them from existing ones, often in the context of algebraic theories and adjunctions, which establish connections between different categories.
Freyd's Adjoint Functor Theorem: Freyd's Adjoint Functor Theorem states that if a functor between categories is a left adjoint, then it preserves all colimits. This theorem provides crucial insights into the relationships between categories, allowing us to understand how structures in one category can be transformed and reflected in another through adjoint functors. The theorem not only highlights the significance of left adjoints in preserving colimits but also plays a vital role in many areas of category theory, including topos theory and homological algebra.
Homology Theory: Homology theory is a mathematical framework that studies topological spaces by associating algebraic structures, typically groups, to them. This theory is used to classify spaces based on their shape and features, providing tools for understanding properties like connectivity and holes in various dimensions. By using functors and natural transformations, homology theory relates different categories, allowing for the exploration of deeper connections within algebraic topology.
Isomorphism: An isomorphism is a special type of morphism in category theory that indicates a structural similarity between two objects, meaning there exists a bijective correspondence between them that preserves the categorical structure. This concept allows us to understand when two mathematical structures can be considered 'the same' in a categorical sense, as it connects to important ideas like special objects, functors, and adjoint relationships.
Left Adjoint: A left adjoint is a functor that, when paired with a right adjoint, establishes a relationship between two categories such that the morphisms in the first category can be 'transferred' to the second category in a way that preserves structure. This concept is pivotal in understanding how different categories interact and allows for the formulation of various important constructions, such as limits and colimits, as well as in defining sheaves and understanding geometric morphisms.
Limit: In category theory, a limit is a universal construction that captures the idea of 'convergence' of objects and morphisms. It formalizes how objects can be combined or related through diagrams, providing a way to describe the most efficient or optimal way to represent a collection of objects and their relationships.
Limit Preservation: Limit preservation refers to the property of a functor that maintains the limits of diagrams when mapping between categories. This means if a diagram has a limit in one category, the functor will map that diagram to another category while preserving the same limit structure. This concept is crucial in understanding adjoint functors, as they often exhibit this behavior and establish deep relationships between different categories.
Natural Transformation: A natural transformation is a way of transforming one functor into another while preserving the structure of the categories involved. It consists of a collection of morphisms that relate the outputs of two functors at each object in the source category, ensuring coherence across all morphisms in that category. This concept links various areas of category theory, such as functor categories and representable functors, through its universal properties and its application in understanding limits and colimits.
Peter Johnstone: Peter Johnstone is a prominent mathematician known for his influential work in category theory and topos theory, which have shaped the understanding of concepts like adjoint functors and subobject classifiers. His contributions provide deep insights into the structure of topoi and their applications in various mathematical fields, including algebraic geometry and logic.
Right adjoint: A right adjoint is a type of functor in category theory that, when paired with a left adjoint, forms an adjunction. In this relationship, the right adjoint maps objects from one category to another while preserving the structure defined by the left adjoint. Understanding right adjoints is crucial as they help define how different categories interact and allow for the translation of concepts between them, particularly in defining unit and counit transformations.
Topos theory: Topos theory is a branch of mathematics that extends set theory concepts to a more abstract context, providing a framework for understanding categories that behave like the category of sets. It allows mathematicians to apply logical and categorical tools to analyze structures in various mathematical disciplines, including geometry, logic, and algebra.
Triangle Identities: Triangle identities are a set of mathematical relationships that express the connections between the angles and sides of triangles. These identities, such as the sine, cosine, and tangent rules, are crucial in various fields including geometry and trigonometry, as they help to solve problems involving triangles by relating the lengths of sides to the measures of angles.
Uniqueness Theorem: The uniqueness theorem in category theory states that certain objects, such as limits, colimits, or exponential objects, are unique up to isomorphism. This means that if an object satisfies specific properties, then it is essentially the same as any other object that also satisfies those properties, providing a powerful tool for establishing equivalences in mathematical structures.
Unit of an adjunction: The unit of an adjunction is a natural transformation that provides a way to relate an object in one category to its image in another category via two functors that form an adjoint pair. This transformation serves as the connecting bridge that helps establish the relationship between the left adjoint functor and the right adjoint functor. It plays a crucial role in understanding how these functors interact and ensures that each object can be mapped appropriately between categories.
Yoneda Lemma: The Yoneda Lemma is a foundational result in category theory that relates functors to natural transformations, stating that every functor from a category to the category of sets can be represented by a set of morphisms from an object in that category. This lemma highlights the importance of morphisms and allows for deep insights into the structure of categories and functors.
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