Shahada ya kwanza (juu)Category TheoryImepangwa
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
Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.
Inajengwa juu ya
Masharti ya awali ya lazima 18 (hadi chini kabisa)
Utajifunza
- 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.
Fasili
- 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.
Nadharia
- 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.
Mawazo potofu ya kawaida
- 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.
Inaelekea kwa
Ugunduzi wa hiari
- ∞-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
Njia za kujifunza
Usomaji zaidi
- Emily Riehl, Category Theory in Context (2016)
- Tom Leinster, Basic Category Theory (2014)
- Saunders Mac Lane, Categories for the Working Mathematician (1971)