本科高年级Category Theory计划中

The Yoneda lemma

The Yoneda lemma, Nat(Hom(A, −), F) ≅ F(A); the full and faithful Yoneda embedding of a category into presheaves; and the principle that an object is determined up to isomorphism by the morphisms into it.

math.cat.yoneda

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前置知识

Representable functors

18 项必需的前置知识（一直追溯到底）

你将学会

State and prove the Yoneda lemma, including its naturality in A and F.
Deduce that the Yoneda embedding C → [Cᵒᵖ, Set] is full and faithful, and that Hom(−, A) ≅ Hom(−, B) implies A ≅ B.
Apply the Yoneda lemma in familiar settings, for example to recover Cayley's theorem for groups.

定义

For a locally small category C, the Yoneda embedding y : C → [Cᵒᵖ, Set] sends A to Hom(−, A).
A presheaf on C is a functor Cᵒᵖ → Set.

定理

Yoneda lemma: for every functor F : C → Set and object A, natural transformations Hom(A, −) ⇒ F correspond bijectively to elements of F(A), via α ↦ α_A(id_A).
The Yoneda embedding is full and faithful; hence A ≅ B whenever Hom(−, A) ≅ Hom(−, B).
Applied to a group regarded as a one-object category, the Yoneda lemma yields Cayley's theorem.
Every presheaf on a small category is a colimit of representable presheaves.

常见误解

Reading Yoneda as saying an object 'is' its set of elements; it says an object is determined by the morphisms into it from all objects.
Expecting the lemma to be hard to prove; its proof is short, and its power lies in how widely it applies.

通向

Elementary topos theory Kan extensions

可选发现

∞-categories An ∞-categorical Yoneda lemma also holds and is a basic tool of higher category theory.
Homotopy type theory In homotopy type theory, path induction can be seen as a type-theoretic analogue of the Yoneda lemma for identity types.
← Products by maps: a first universal property
← Functors
← Adjunctions

学习路径

Research preparation