The Yoneda Lemma is a powerful tool in category theory with wide-ranging applications. It establishes a deep connection between objects and their representable functors, allowing us to study complex structures through simpler functorial representations.

From algebraic geometry to sheaf theory, the Yoneda Lemma provides a unifying framework for understanding diverse mathematical concepts. It's crucial for proving the existence of adjoint functors and plays a key role in the theory of Kan extensions.

Yoneda Lemma Applications in Category Theory

Applications in algebraic geometry

• Yoneda lemma establishes a connection between an object and its representable functor
  • For an object $A$ in a category $\mathcal{C}$, the representable functor $\text{Hom}_{\mathcal{C}}(A, -)$ is fully faithful captures all the information about the object $A$
  • Allows studying objects in a category through their functors provides a powerful tool for understanding the structure of objects
• In algebraic geometry, schemes can be studied via their functors of points
  • A scheme $X$ represented by the functor $h_X : \text{Sch}^{\text{op}} \to \text{Set}$, where $h_X(S) = \text{Hom}_{\text{Sch}}(S, X)$ associates to each scheme $S$ the set of morphisms from $S$ to $X$
  • Yoneda lemma ensures that this functor fully captures the scheme $X$ allows for a complete understanding of the scheme through its functor of points
• Yoneda lemma used to prove the existence of the Hilbert scheme parametrizes closed subschemes of a given scheme with a fixed Hilbert polynomial
  • Hilbert functor $\text{Hilb}_{P(t)}(X) : \text{Sch}^{\text{op}} \to \text{Set}$ associates to each scheme $S$ the set of closed subschemes of $X \times S$ with Hilbert polynomial $P(t)$
  • If this functor is representable, the representing object is the Hilbert scheme by the Yoneda lemma

Proofs for adjoint functors

• Yoneda lemma is a powerful tool for proving the existence of adjoint functors establishes a correspondence between certain functors and their adjoints
• Given functors $F : \mathcal{C} \to \mathcal{D}$ and $G : \mathcal{D} \to \mathcal{C}$, $F$ is left adjoint to $G$ (and $G$ is right adjoint to $F$) if there is a natural isomorphism:
  • $\text{Hom}{\mathcal{D}}(F(C), D) \cong \text{Hom}{\mathcal{C}}(C, G(D))$ for all objects $C$ in $\mathcal{C}$ and $D$ in $\mathcal{D}$ establishes a bijection between morphisms in the two categories
• To prove the existence of a left adjoint to a functor $G$, it suffices to show that the functor $\text{Hom}_{\mathcal{D}}(-, G(-))$ is representable
  • By the Yoneda lemma, the representing object of this functor is the left adjoint of $G$ provides a concrete way to construct the adjoint functor
• Dually, to prove the existence of a right adjoint to a functor $F$, it suffices to show that the functor $\text{Hom}_{\mathcal{C}}(F(-), -)$ is representable by the Yoneda lemma

Role in sheaves and stacks

• Yoneda lemma is fundamental in the theory of sheaves and stacks provides a way to study geometric objects through their functors
• A presheaf on a topological space $X$ is a contravariant functor $F : \text{Open}(X)^{\text{op}} \to \text{Set}$ associates to each open set of $X$ a set, and to each inclusion of open sets a restriction map
  • The category of presheaves on $X$ is denoted by $\text{PSh}(X)$ is an important object of study in sheaf theory
• A sheaf is a presheaf satisfying the gluing axiom allows for the construction of global sections from local data
  • The category of sheaves on $X$ is denoted by $\text{Sh}(X)$ is a full subcategory of $\text{PSh}(X)$
• Yoneda lemma implies that the Yoneda embedding $y : \text{Open}(X) \to \text{PSh}(X)$, given by $y(U) = \text{Hom}_{\text{Open}(X)}(-, U)$, is fully faithful
  • Allows for studying open sets of $X$ through their representable presheaves provides a way to understand the topology of $X$ through functors
• A stack is a generalization of a sheaf, where the gluing axiom is replaced by a weaker condition involving groupoids allows for the study of geometric objects with automorphisms
  • Yoneda lemma plays a crucial role in the theory of stacks, as it allows for studying geometric objects through their functors of points provides a powerful tool for understanding moduli problems

Connection to Kan extensions

• Yoneda lemma is closely related to the theory of Kan extensions provides a way to extend functors along other functors
• Given functors $F : \mathcal{C} \to \mathcal{D}$ and $K : \mathcal{C} \to \mathcal{E}$, a left Kan extension of $F$ along $K$ is a functor $\text{Lan}_K F : \mathcal{E} \to \mathcal{D}$ together with a natural transformation $\eta : F \Rightarrow \text{Lan}_K F \circ K$ satisfying a universal property
• Yoneda lemma can be used to prove the existence of left Kan extensions
  • Left Kan extension of $F$ along $K$ can be computed as $(\text{Lan}K F)(E) = \int^{C \in \mathcal{C}} \text{Hom}{\mathcal{E}}(K(C), E) \cdot F(C)$ is a weighted colimit
  • Yoneda lemma ensures that this formula defines a functor $\text{Lan}_K F : \mathcal{E} \to \mathcal{D}$ satisfying the required universal property
• Dually, Yoneda lemma can be used to prove the existence of right Kan extensions
  • Right Kan extension of $F$ along $K$ can be computed as $(\text{Ran}K F)(E) = \int{C \in \mathcal{C}} \text{Hom}_{\mathcal{E}}(E, K(C)) \pitchfork F(C)$ is a weighted limit
  • Yoneda lemma ensures that this formula defines a functor $\text{Ran}_K F : \mathcal{E} \to \mathcal{D}$ satisfying the required universal property