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Yoneda LemmaEdit

The Yoneda Lemma is a foundational result in category theory that explains how an object of a category can be understood entirely by the way maps into it vary across the whole category. Named after the Japanese mathematician Noboru Yoneda, it provides a precise bridge between objects and the presheaves they represent. The lemma underpins the canonical embedding of a category into a world of set-valued functors and serves as a guiding principle in many areas of mathematics, from algebra to geometry to topology.

Statement of the Yoneda Lemma

Let C be a locally small category, let A be an object of C, and let F: C^op -> Set be a functor (a presheaf on C). The Yoneda Lemma asserts that there is a natural bijection

Nat(Hom(-, A), F) ≅ F(A),

where Nat denotes the set of natural transformations and Hom(-, A) is the representable presheaf that assigns to each object X in C the set C(X, A). This bijection is natural in both the variable object A and the functor F.

Two things to notice in this statement:

  • The left-hand side uses the contravariant hom-functor Hom(-, A): for each X in C, it gives the set of morphisms from X to A; a natural transformation from this presheaf to F is a coherent way to pick an element of F(X) for every X, compatible with maps in C.
  • The bijection is natural in A and F, meaning it respects morphisms in C and natural transformations between functors.

The proof constructs the correspondence by evaluating a natural transformation at the identity morphism id_A: A -> A. Every morphism f: X -> A yields an induced component of the natural transformation, and conversely, any element of F(A) determines a natural transformation by precomposing with morphisms into A. The result can be summarized by saying that the object A is faithfully represented by the presheaf Hom(-, A), a theme that recurs throughout categorical practice.

Consequences and Interpretations

  • Yoneda embedding: The assignment A ↦ Hom(-, A) defines a functor j: C -> Set^[C^op], sending each object to its representable presheaf. This is known as the Yoneda embedding and is full and faithful, so C can be viewed as a subcategory of the presheaf category [[[C]^op, Set]]. This perspective often simplifies reasoning about morphisms by moving them into a functorial setting.
  • Representation and reconstruction: The lemma shows that an object is determined, up to isomorphism, by the way maps into it interact with all other objects in C. In particular, when F is itself a representable presheaf, the Yoneda Lemma recovers familiar hom-set isomorphisms such as Nat(Hom(-, A), Hom(-, B)) ≅ Hom(A, B).
  • Applicability across mathematics: Because presheaves arise naturally in many areas (e.g., algebraic geometry, topology, and sheaf theory), the Yoneda Lemma provides a unifying tool for translating questions about objects into questions about sets of maps and their coherence. See for instance how it informs constructions in algebraic geometry and topology.

Representable Functors and the Yoneda Embedding

  • A presheaf is representable if it is naturally isomorphic to Hom(-, A) for some A in C. The Yoneda Lemma explains why representable presheaves are so central: they classify objects up to the lens of their maps.
  • The Yoneda embedding j: C -> Set^[C^op] is not only faithful but full, meaning that every natural transformation between representables is induced by a unique morphism in C. This observation often reduces problems about morphisms in C to problems about natural transformations between presheaves.
  • The lemma also has a covariant counterpart for functors of the form F: C -> Set and the contravariant version, highlighting the dual nature of hom-sets with respect to the direction of morphisms.

Variants and Generalizations

  • Enriched Yoneda Lemma: In an enriched setting, where hom-objects take values in a monoidal category V rather than in Set, the lemma extends to show a bijection between enriched natural transformations and enriched hom-objects. This generalization is central to enriched category theory and underpins many constructions in stable homotopy theory and algebraic geometry.
  • Presheaves vs copresheaves: The standard Yoneda Lemma uses presheaves (contravariant set-valued functors). A dual statement holds for copresheaves (covariant set-valued functors) with appropriate adjustments to the hom-functor.
  • Higher and derived versions: In more advanced contexts, one can consider Yoneda-type statements in higher categories, ∞-categories, or in topos theory, where the ideas about representing objects via natural transformations persist in more flexible settings.

Historical remarks

Noboru Yoneda formulated the lemma in the 1950s, and it quickly became a cornerstone of modern category theory. The idea that objects could be studied through their maps into or out of them revolutionized the way mathematicians think about structure and relationships. Over time, the lemma has found applications in a wide range of disciplines, including algebraic geometry, homological algebra, and sheaf theory.

See also