Glossary

8.3 Kan extensions: left and right

3 min read•july 23, 2024

Kan extensions are powerful tools in category theory, allowing us to extend functors along other functors. They come in two flavors: left and right, each with unique universal properties that make them useful in various mathematical contexts.

From theory to vector spaces, Kan extensions have wide-ranging applications. They're closely tied to adjoint functors and play crucial roles in homological algebra, algebraic geometry, and representation theory, helping us understand complex mathematical relationships.

Kan Extensions

Definition of Kan extensions

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    • Extends a functor F:C→DF: \mathcal{C} \to \mathcal{D} along another functor K:C→EK: \mathcal{C} \to \mathcal{E} to a functor [Lan_K F](https://www.fiveableKeyTerm:lan_k_f): \mathcal{E} \to \mathcal{D}
    • Satisfies a for any other functor G:E→DG: \mathcal{E} \to \mathcal{D} with a from to the composition of and KK
      • There exists a unique natural transformation β\beta from LanKFLan_K F to GG such that the composition of β\beta and the unit of the Kan extension equals the original natural transformation
    • Extends a functor F:C→DF: \mathcal{C} \to \mathcal{D} along another functor K:C→EK: \mathcal{C} \to \mathcal{E} to a functor [Ran_K F](https://www.fiveableKeyTerm:ran_k_f): \mathcal{E} \to \mathcal{D}
    • Satisfies a universal property for any other functor G:E→DG: \mathcal{E} \to \mathcal{D} with a natural transformation from the composition of GG and KK to FF
      • There exists a unique natural transformation β\beta from GG to RanKFRan_K F such that the composition of the counit of the Kan extension and β\beta equals the original natural transformation

Construction in familiar categories

  • In the category of sets (Set)
    • Left Kan extension LanKFLan_K F computes the coproduct (disjoint union) of F(c)F(c) over all objects cc in C\mathcal{C} that map to a given object ee in E\mathcal{E} under KK
      • (LanKF)(e)=∐c∈C,K(c)→eF(c)(Lan_K F)(e) = \coprod_{c \in \mathcal{C}, K(c) \to e} F(c)
    • Right Kan extension RanKFRan_K F computes the product of F(c)F(c) over all objects cc in C\mathcal{C} that a given object ee in E\mathcal{E} maps to under KK
      • (RanKF)(e)=∏c∈C,e→K(c)F(c)(Ran_K F)(e) = \prod_{c \in \mathcal{C}, e \to K(c)} F(c)
  • In the category of vector spaces ()
    • Left Kan extension LanKFLan_K F computes the direct sum of F(c)F(c) over all objects cc in C\mathcal{C} that map to a given object ee in E\mathcal{E} under KK
      • (LanKF)(e)=⨁c∈C,K(c)→eF(c)(Lan_K F)(e) = \bigoplus_{c \in \mathcal{C}, K(c) \to e} F(c)
    • Right Kan extension RanKFRan_K F computes the direct product of F(c)F(c) over all objects cc in C\mathcal{C} that a given object ee in E\mathcal{E} maps to under KK
      • (RanKF)(e)=∏c∈C,e→K(c)F(c)(Ran_K F)(e) = \prod_{c \in \mathcal{C}, e \to K(c)} F(c)

Relationship to adjoint functors

  • Adjoint functors and Kan extensions
    • For an adjoint pair of functors (F,G)(F, G) with F:C→DF: \mathcal{C} \to \mathcal{D} and G:D→CG: \mathcal{D} \to \mathcal{C}
      • FF is the left Kan extension of the composition F∘GF \circ G along GG
      • GG is the right Kan extension of the composition G∘FG \circ F along FF
  • Unit and counit of an
    • The unit η:1C⇒G∘F\eta: 1_\mathcal{C} \Rightarrow G \circ F corresponds to the unit of the left Kan extension of F∘GF \circ G along GG
    • The counit ε:F∘G⇒1D\varepsilon: F \circ G \Rightarrow 1_\mathcal{D} corresponds to the counit of the right Kan extension of G∘FG \circ F along FF

Applications in mathematical contexts

  • Kan extensions in homological algebra
    • Derived functors (Tor and Ext) can be expressed as Kan extensions of tensor product and Hom functors along suitable functors between categories of modules
      • Allows for the computation of derived functors in a more general setting
  • Kan extensions in algebraic geometry
    • Pushforward and pullback functors between categories of sheaves can be described as Kan extensions along suitable functors between the underlying topological spaces
      • Provides a way to relate sheaves on different spaces
  • Kan extensions in representation theory
    • Induction and restriction functors between categories of representations can be expressed as Kan extensions along suitable functors between the underlying groups or algebras
      • Enables the study of representations of related groups or algebras

Key Terms to Review (17)

$f$: $f$ typically represents a morphism or arrow between two objects in a category. In the context of Kan extensions, $f$ serves as a critical mapping that connects the domain and codomain while facilitating the extension process, whether it's a left or right Kan extension. Understanding the role of $f$ is essential for grasping how these extensions generalize and preserve properties within categories.
$g$: $g$ is a functor that represents a specific morphism in the context of Kan extensions, acting as a bridge between two categories by facilitating the transfer of structures and properties. This morphism plays a crucial role in defining both left and right Kan extensions, which are important concepts in category theory that generalize how we can extend functors while preserving certain limits or colimits.
Adjoint Functor Theorem: The adjoint functor theorem establishes a deep connection between categories, providing conditions under which functors can be considered left or right adjoints. This theorem is crucial for understanding the nature of relationships between different mathematical structures and helps in characterizing properties like limits and colimits in terms of adjunctions. Its implications are felt across various topics, linking concepts like equivalence of categories and the properties of functors that preserve certain structures.
Adjunction: Adjunction is a fundamental concept in category theory where two functors stand in a specific relationship, known as an adjoint pair. This relationship allows one functor to be thought of as providing a best approximation of the other, capturing the essence of universal properties. Adjunctions are critical in many areas of mathematics, including the study of limits, colimits, and various constructions that connect different categories.
Categorical Colimits: Categorical colimits are a way to generalize the notion of combining objects in a category into a single object that summarizes the information from those objects. They are defined through a universal property that captures the idea of 'gluing together' diagrams of objects and morphisms, allowing for coherent constructions that maintain the relationships between the original elements. Colimits include various constructions such as coproducts, coequalizers, and pushouts, playing a critical role in understanding how structures can be built or extended in categorical contexts.
Categorical limits: Categorical limits are a fundamental concept in category theory that generalizes the idea of limits in mathematics, capturing how diagrams (collections of objects and morphisms) behave in a category. They represent a way to construct a universal object from a diagram, providing a solution that is unique up to isomorphism and fulfilling specific properties, such as being a cone over the diagram. This concept connects deeply with Kan extensions, which extend functors between categories and involve limits when defining the left and right extensions.
Contravariant Functor: A contravariant functor is a type of functor that reverses the direction of morphisms when mapping between categories. Instead of mapping arrows in a category to arrows in another category directly, it maps arrows in the opposite direction, reflecting a form of duality that has important implications in various areas of mathematics.
Covariant Functor: A covariant functor is a mapping between categories that preserves the structure of the categories by associating each object in one category to an object in another category and each morphism in the first category to a morphism in the second category, maintaining the direction of morphisms. This concept is fundamental as it allows for the systematic translation of structures and relationships from one context to another, helping to illustrate connections across different mathematical frameworks.
Functoriality: Functoriality refers to the principle that a functor preserves the structure of categories by mapping objects and morphisms from one category to another in a way that respects the composition of morphisms and identity morphisms. This concept is foundational in category theory, allowing for the transformation and comparison of mathematical structures while maintaining their essential properties.
Lan_k f: The term $\text{lan}_k f$ refers to the left Kan extension of a functor $f$ along a functor $k$. It represents a way of extending the domain of a functor while preserving certain structures, specifically by providing the best possible approximation of $f$ in terms of the functor $k$. This concept is crucial in category theory as it allows for a systematic method to derive new functors from existing ones, facilitating the understanding of relationships between different categories.
Left Kan Extension: The left Kan extension is a functorial construction that allows you to extend a functor defined on a category to a larger category in a way that preserves certain properties. This construction provides a means to create new functors from existing ones while maintaining the structure and relationships of the original categories involved. It is important in various areas of category theory, especially in understanding natural transformations and adjunctions, and has significant implications for applications of the Yoneda lemma.
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 family of morphisms that connect the objects in one category to their images in another category, ensuring that the relationships between the objects are maintained across different mappings. This concept ties together various important aspects of category theory, allowing mathematicians to relate different structures in a coherent manner.
Ran_k f: The term $\text{ran}_k \text{f}$, or the $k$-th range of a functor $f$, refers to the image of the functor when applied to morphisms in a category. It helps to understand how a functor acts on objects and morphisms within a certain context, particularly when analyzing Kan extensions, which involve the extension of a functor along another functor. This concept plays a vital role in determining how certain structures can be constructed or represented based on existing ones.
Right Kan Extension: Right Kan extension is a way to extend a functor defined on a small category to a functor on a larger category, ensuring that it respects certain universal properties. This concept is crucial in category theory, particularly when discussing the relationship between functors and natural transformations, as it provides a means of 'lifting' data while preserving structure. Right Kan extensions are instrumental in various applications, such as those involving the Yoneda lemma and in characterizing adjoint functors.
Set: A set is a well-defined collection of distinct objects, considered as an object in its own right. In category theory, sets serve as the fundamental building blocks for constructing more complex mathematical structures, allowing for the exploration of relationships and mappings between different sets through functions.
Universal Property: A universal property is a fundamental concept in category theory that describes an object in terms of its relationships with other objects through morphisms. It serves as a characterization of objects that can uniquely determine them via certain properties, often in the context of limits and colimits, making them essential for understanding constructions like products, coproducts, and adjoint functors.
Vect: In category theory, 'vect' typically refers to the category of vector spaces over a field, with linear transformations as morphisms. This category is fundamental in understanding structures in mathematics where linearity is a key feature, allowing for connections to various concepts such as functors, adjunctions, and monoidal categories.
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