4.3 Natural transformations and functor categories
3 min read•august 7, 2024
Natural transformations are the secret sauce of functors, letting us compare them smoothly. They're like bridges between functors, with components that play nice with morphisms. Think of them as the glue that holds the functor world together.
Functor categories take things up a notch, treating functors as objects and natural transformations as morphisms. It's like a meta-category where functors themselves become the stars of the show. This setup lets us study functors in a whole new light.
Natural Transformations
Definition and Components
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η:F→G consists of a family of morphisms ηX:F(X)→G(X) for each object X in the source category
Components of a natural transformation are the individual morphisms ηX that make up the natural transformation
Components must satisfy the for every f:X→Y in the source category: G(f)∘ηX=ηY∘F(f)
Naturality condition ensures that the components of the natural transformation are compatible with the morphisms in the source and target categories
Commutative diagram can be used to visualize the naturality condition
Natural Isomorphisms and Properties
is a natural transformation η:F→G where each component ηX is an in the target category
Inverse of a natural isomorphism is also a natural transformation, denoted as η−1:G→F
Identity natural transformation 1F:F→F has components (1F)X=1F(X) for each object X in the source category
Composition of natural transformations η:F→G and μ:G→H is defined componentwise: (μ∘η)X=μX∘ηX
Natural transformations can be used to define (if there exist natural isomorphisms between two functors)
Functor Categories
Definition and Objects
CD has functors F:D→C as objects
Morphisms in the functor category are natural transformations between the functors
Identity functor 1D:D→D serves as the identity object in the functor category
Constant functor ΔX:D→C sends every object in D to a fixed object X in C and every morphism in D to the identity morphism 1X
Composition of Natural Transformations
Vertical composition of natural transformations η:F→G and μ:G→H is defined as μ∘η:F→H
Horizontal composition of natural transformations η:F→G in CD and θ:H→K in DE is a natural transformation η∗θ:F∘H→G∘K in CE
Horizontal composition is defined componentwise: (η∗θ)X=ηK(X)∘F(θX)
Interchange law holds for vertical and horizontal composition: (μ∘η)∗(θ∘ϕ)=(μ∗θ)∘(η∗ϕ)
Yoneda Lemma
Statement and Consequences
states that for any functor F:Cop→Set and any object X in C, there is a bijection between the set of natural transformations Nat(HomC(X,−),F) and the set F(X)
Bijection is given by evaluating a natural transformation at the identity morphism of X
Yoneda lemma implies that the functor HomC(X,−) (represented functor) completely determines the object X up to isomorphism
Yoneda embedding C→SetCop sends each object X to its represented functor HomC(X,−) and each morphism f:X→Y to the natural transformation HomC(f,−)
Yoneda embedding is fully faithful, meaning that it preserves and reflects isomorphisms between objects and morphisms