2.3 Functor categories and the Yoneda lemma

Functor categories organize functors and natural transformations, providing a powerful framework for studying relationships between categories. They allow us to view functors as objects and natural transformations as morphisms, creating a rich structure for analysis.

The Yoneda lemma is a cornerstone of category theory, establishing a deep connection between objects and their relationships. It reveals how objects are determined by their interactions within a category, offering new perspectives on categorical structures and their properties.

Functor Categories

Functor categories and components

• Functor category organizes functors between categories C and D as objects and natural transformations as morphisms denoted as $[C, D]$ or $D^C$
• Objects represent functors $F: C \to D$ mapping category C to category D (homomorphisms)
• Morphisms consist of natural transformations $\alpha: F \to G$ between functors F and G (structure-preserving maps)
• Composition follows vertical composition of natural transformations preserving functor properties
• Identity morphism corresponds to identity natural transformation for each functor maintaining structure

Yoneda lemma and interpretations

• Yoneda lemma establishes natural bijection $\text{Nat}(y(c), F) \cong F(c)$ for locally small category C, object $c \in C$, and functor $F: C \to \text{Set}$
• Interpretation reveals one-to-one correspondence between natural transformations from representable functors to F and elements of F(c)
• Objects in category determined by relationships to other objects through morphisms
• Allows studying objects through interactions within category structure
• Provides embedding of any category into category of functors expanding analytical tools

Applications of Yoneda lemma

• Representable functors take form $\text{Hom}(c, -)$ for object c in category
• Yoneda lemma identifies representable functors by showing natural isomorphisms
• Universal elements correspond to natural isomorphisms between functor and representable functor
• Proves properties of adjoint functors using Yoneda lemma's bijection
• Demonstrates representable functors preserve limits enhancing categorical analysis

Yoneda embedding properties

• Yoneda embedding defined as functor $y: C \to [C^{op}, \text{Set}]$ with $y(c) = \text{Hom}(-, c)$
• Fully faithful proof shows bijective map $\text{Hom}_C(a, b) \to \text{Hom}_{[C^{op}, \text{Set}]}(y(a), y(b))$ for any $a, b \in C$
• Consequences include embedding C into category of presheaves on C
• Every category viewed as full subcategory of functor category expanding analytical scope
• Yoneda embedding preserves all limits existing in C maintaining structural properties
• Density theorem states every functor $F: C^{op} \to \text{Set}$ as colimit of representable functors generalizing representation theory