2.2 Functors and natural transformations

Functors and natural transformations are the backbone of category theory. They allow us to map between categories, preserving their structure, and compare different functors. These concepts are crucial for understanding how mathematical structures relate to each other.

In this section, we'll dive into the nitty-gritty of functors and natural transformations. We'll look at their definitions, properties, and how they're used in various mathematical contexts. This knowledge will help us grasp the power of categorical thinking.

Functors: Structure-Preserving Maps

Definition and Types of Functors

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• Functors map between categories preserving categorical structure (objects, morphisms, relationships)
• Covariant functors preserve morphism direction while contravariant functors reverse it
• Forgetful functor maps from categories with more structure to less (groups to sets)
• Power set functor maps category of sets to itself (sets to power sets, functions to image functions)

Examples in Mathematical Contexts

• Fundamental group functor in algebraic topology assigns topological spaces to fundamental groups and continuous maps to induced homomorphisms
• Yoneda embedding embeds locally small categories into categories of presheaves
• Functors connect abstract algebraic structures to concrete linear algebraic objects in representation theory

Properties of Functors

Action on Objects and Morphisms

• Functor F: C → D assigns objects X in C to F(X) in D and morphisms f: X → Y in C to F(f): F(X) → F(Y) in D
• Preserves identity morphisms F(idX) = idF(X) for any object X in C
• Preserves composition F(g ∘ f) = F(g) ∘ F(f) for morphisms f: X → Y and g: Y → Z in C
• Action must be well-defined (unique object or morphism in codomain for each input)

Functor Classifications

• Faithful functors inject on morphisms
• Full functors surject on morphisms
• Fully faithful functors biject on morphisms
• Equivalence of categories fully faithful and essentially surjective on objects
• Adjoint functors describe special relationships between functor pairs in opposite directions (natural bijections between morphism sets)

Natural Transformations: Morphisms Between Functors

Definition and Properties

• Natural transformation α: F ⇒ G between functors F, G: C → D family of morphisms αX: F(X) → G(X) in D for each object X in C
• Satisfies naturality condition G(f) ∘ αX = αY ∘ F(f) for any morphism f: X → Y in C
• Compares functors and expresses relationships between functorial constructions
• Isomorphism of functors natural transformation with two-sided inverse (functors essentially the same)

Applications in Category Theory

• Crucial in defining adjunctions between categories (unit and counit transformations)
• Yoneda lemma expressed using natural transformations provides insights into category structure
• Formulates coherence conditions in categorical structures (monoidal categories, 2-categories)

Composition of Functors and Natural Transformations

Functor Composition

• Composition of F: C → D and G: D → E defined as G ∘ F: C → E
• (G ∘ F)(X) = G(F(X)) for objects
• (G ∘ F)(f) = G(F(f)) for morphisms

Natural Transformation Composition

• Vertical composition of α: F ⇒ G and β: G ⇒ H yields β ∘ α: F ⇒ H
• (β ∘ α)X = βX ∘ αX defined componentwise
• Horizontal composition of α: F ⇒ G and β: H ⇒ K yields β * α: F ∘ H ⇒ G ∘ K
• Satisfies coherence conditions

Functor Categories

• Fun(C, D) has functors from C to D as objects and natural transformations as morphisms
• Yoneda embedding fully faithful functor from C to Fun(Cop, Set)
• Provides natural setting for representation theory (representations as functors)
• 2-category Cat has categories as objects, functors as 1-morphisms, natural transformations as 2-morphisms
• Illustrates hierarchical nature of categorical structures