11.2 Functors and Natural Transformations

Functors and natural transformations are the building blocks of category theory. They allow us to map between categories and compare different functors, giving us powerful tools to analyze mathematical structures and relationships.

These concepts help us understand how different mathematical objects relate to each other. By using functors and natural transformations, we can uncover deep connections between seemingly unrelated areas of math, leading to new insights and discoveries.

Functors and their properties

Definition and structure of functors

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• A functor is a structure-preserving map between categories, consisting of an object map and a morphism map
• The object map assigns to each object in the source category an object in the target category
• The morphism map assigns to each morphism in the source category a morphism in the target category
  • Preserves the domain and codomain of the morphisms (if $f: A \to B$ in the source category, then $F(f): F(A) \to F(B)$ in the target category)

Properties of functors

• Functors preserve identity morphisms
  • The image of an identity morphism under a functor is an identity morphism (if $id_A$ is the identity morphism of object $A$ in the source category, then $F(id_A) = id_{F(A)}$ in the target category)
• Functors preserve composition
  • The image of a composite morphism under a functor is equal to the composition of the images of the individual morphisms (if $f: A \to B$ and $g: B \to C$ in the source category, then $F(g \circ f) = F(g) \circ F(f)$ in the target category)
• Examples of functors
  • The identity functor $Id_C: C \to C$ maps each object and morphism in a category $C$ to itself
  • The forgetful functor $U: Grp \to Set$ maps each group to its underlying set and each group homomorphism to its underlying function

Composition of functors

Composing functors

• Functors can be composed, forming a new functor from the target category of the first functor to the target category of the second functor
  • If $F: C \to D$ and $G: D \to E$ are functors, then their composition $G \circ F: C \to E$ is a functor
  • The composition of functors is defined by $(G \circ F)(A) = G(F(A))$ for objects and $(G \circ F)(f) = G(F(f))$ for morphisms

Properties of functor composition

• The composition of functors is associative
  • $(F \circ G) \circ H = F \circ (G \circ H)$ for functors $F$, $G$, and $H$
• The identity functor acts as the identity element for functor composition
  • $Id_C \circ F = F = F \circ Id_D$ for a functor $F: C \to D$
• The composition of functors preserves the properties of functors
  • Such as the preservation of identity morphisms and composition
• Example of functor composition
  • Consider the functors $F: Set \to Grp$ (free group functor) and $U: Grp \to Set$ (forgetful functor). Their composition $U \circ F: Set \to Set$ maps each set to its underlying set of the free group generated by that set

Natural transformations in category theory

Definition and components of natural transformations

• A natural transformation is a morphism between functors, providing a way to compare and relate different functors between the same categories
• Given two functors $F, G: C \to D$, a natural transformation $\eta: F \Rightarrow G$ assigns to each object $X$ in $C$ a morphism $\eta_X: F(X) \to G(X)$ in $D$
  • Such that for every morphism $f: X \to Y$ in $C$, the following diagram commutes: $G(f) \circ \eta_X = \eta_Y \circ F(f)$
• The components of a natural transformation, $\eta_X$, are the morphisms assigned to each object $X$ in the source category

Composition and properties of natural transformations

• Natural transformations can be composed vertically
  • When the target functor of one natural transformation is the source functor of another (if $\eta: F \Rightarrow G$ and $\theta: G \Rightarrow H$, then $\theta \circ \eta: F \Rightarrow H$ is a natural transformation)
• Natural transformations can be composed horizontally
  • When the functors involved have a common source or target category (if $\eta: F \Rightarrow G$ and $\theta: H \Rightarrow K$, then $\eta \times \theta: F \circ H \Rightarrow G \circ K$ is a natural transformation)
• Natural transformations provide a way to study the relationships between functors and to define equivalences between categories
  • Two categories $C$ and $D$ are equivalent if there exist functors $F: C \to D$ and $G: D \to C$ such that $G \circ F \cong Id_C$ and $F \circ G \cong Id_D$ via natural isomorphisms
• Example of a natural transformation
  • The determinant $det: GL_n \Rightarrow \mathbb{R}^*$ is a natural transformation from the general linear group functor $GL_n: Ring \to Grp$ to the multiplicative group functor $\mathbb{R}^*: Ring \to Grp$

Functors vs Natural transformations

Functors and natural transformations form a 2-category

• Categories as objects, functors as morphisms, and natural transformations as 2-morphisms
• The functor category $[C, D]$ has functors from $C$ to $D$ as objects and natural transformations between these functors as morphisms

Adjoint functors and universal properties

• Adjoint functors are pairs of functors $(F: C \to D, G: D \to C)$ with a natural isomorphism between the hom-sets $Hom_D(F(X), Y)$ and $Hom_C(X, G(Y))$
• Adjoint functors play a crucial role in category theory and are closely related to universal properties
  • Universal properties characterize objects and morphisms in terms of their relationships with other objects and morphisms in the category
  • Examples of adjoint functors: free and forgetful functors, product and diagonal functors, and exponential and evaluation functors

The Yoneda lemma and its implications

• The Yoneda lemma establishes a relationship between an object and its representable functor
  • A representable functor $Hom_C(X, -): C \to Set$ assigns to each object $Y$ in $C$ the set of morphisms $Hom_C(X, Y)$
• The Yoneda lemma states that for any functor $F: C \to Set$ and object $X$ in $C$, there is a bijection between the set of natural transformations $Nat(Hom_C(X, -), F)$ and the set $F(X)$
• The Yoneda lemma is a fundamental result in category theory that involves functors and natural transformations
  • It allows for the study of objects in a category through their relationships with other objects, as captured by the representable functors

Applications of functors and natural transformations

• Functors and natural transformations are used to study and compare different mathematical structures
  • Such as algebras, topological spaces, and sheaves, by examining their categorical properties and relationships
• Functors and natural transformations provide a unified framework for understanding and relating various mathematical concepts
  • For example, the concept of a group can be generalized to the notion of a group object in a category, which is defined using functors and natural transformations
• Functors and natural transformations are essential tools in homological algebra and algebraic topology
  • They allow for the construction of algebraic invariants (such as homology and cohomology groups) that capture important topological properties of spaces