🔢Category Theory Unit 8 Review

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 set 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

Left Kan extension
- Extends a functor $F: \mathcal{C} \to \mathcal{D}$ along another functor $K: \mathcal{C} \to \mathcal{E}$ to a functor $Lan_K F: \mathcal{E} \to \mathcal{D}$
- Satisfies a universal property for any other functor $G: \mathcal{E} \to \mathcal{D}$ $G : E \to D$ with a natural transformation from $F$ $F$ to the composition of $G$ $G$ and $K$ $K$
  - There exists a unique natural transformation $\beta$ from $Lan_K F$ to $G$ such that the composition of $\beta$ and the unit of the Kan extension equals the original natural transformation
Right Kan extension
- Extends a functor $F: \mathcal{C} \to \mathcal{D}$ along another functor $K: \mathcal{C} \to \mathcal{E}$ to a functor $Ran_K F: \mathcal{E} \to \mathcal{D}$
- Satisfies a universal property for any other functor $G: \mathcal{E} \to \mathcal{D}$ $G : E \to D$ with a natural transformation from the composition of $G$ $G$ and $K$ $K$ to $F$ $F$
  - There exists a unique natural transformation $\beta$ from $G$ to $Ran_K F$ such that the composition of the counit of the Kan extension and $\beta$ equals the original natural transformation

Construction in familiar categories

In the category of sets (Set)
- Left Kan extension $Lan_K F$ $L a n_{K} F$ computes the coproduct (disjoint union) of $F(c)$ $F (c)$ over all objects $c$ $c$ in $\mathcal{C}$ $C$ that map to a given object $e$ $e$ in $\mathcal{E}$ $E$ under $K$ $K$
  - $(Lan_K F)(e) = \coprod_{c \in \mathcal{C}, K(c) \to e} F(c)$
- Right Kan extension $Ran_K F$ $R a n_{K} F$ computes the product of $F(c)$ $F (c)$ over all objects $c$ $c$ in $\mathcal{C}$ $C$ that a given object $e$ $e$ in $\mathcal{E}$ $E$ maps to under $K$ $K$
  - $(Ran_K F)(e) = \prod_{c \in \mathcal{C}, e \to K(c)} F(c)$
In the category of vector spaces (Vect)
- Left Kan extension $Lan_K F$ $L a n_{K} F$ computes the direct sum of $F(c)$ $F (c)$ over all objects $c$ $c$ in $\mathcal{C}$ $C$ that map to a given object $e$ $e$ in $\mathcal{E}$ $E$ under $K$ $K$
  - $(Lan_K F)(e) = \bigoplus_{c \in \mathcal{C}, K(c) \to e} F(c)$
- Right Kan extension $Ran_K F$ $R a n_{K} F$ computes the direct product of $F(c)$ $F (c)$ over all objects $c$ $c$ in $\mathcal{C}$ $C$ that a given object $e$ $e$ in $\mathcal{E}$ $E$ maps to under $K$ $K$
  - $(Ran_K F)(e) = \prod_{c \in \mathcal{C}, e \to K(c)} F(c)$

Definition of Kan extensions, Two Triangle Kanbans | AllAboutLean.com

Relationship to adjoint functors

Adjoint functors and Kan extensions
- For an adjoint pair of functors $(F, G)$ $(F, G)$ with $F: \mathcal{C} \to \mathcal{D}$ $F : C \to D$ and $G: \mathcal{D} \to \mathcal{C}$ $G : D \to C$
  - $F$ is the left Kan extension of the composition $F \circ G$ along $G$
  - $G$ is the right Kan extension of the composition $G \circ F$ along $F$
Unit and counit of an adjunction
- The unit $\eta: 1_\mathcal{C} \Rightarrow G \circ F$ corresponds to the unit of the left Kan extension of $F \circ G$ along $G$
- The counit $\varepsilon: F \circ G \Rightarrow 1_\mathcal{D}$ corresponds to the counit of the right Kan extension of $G \circ F$ along $F$

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

🔢Category Theory Unit 8 Review