Lecture 15 L15 164 Presheaf categories op Let C be a small category.ThefunctorcategorySetC is called the category of preseaves on C. ! objects are contravariant functors from C to Set ! morphisms are natural transformations Much used in the semantics of various dependently-typed languages and logics. L15 165 Yoneda functor op y:C → SetC (where C is a small category) is the Curried version of the hom functor Hom C × Cop =∼ Cop × C −→C Set L15 166 Yoneda functor op y:C → SetC (where C is a small category) is the Curried version of the hom functor Hom C × Cop =∼ Cop × C −→C Set op ! For each C-object X,theobjectyX ∈ SetC is the functor C(_ , X) : Cop → Set given by Z &→ C(Z, X) g ◦ f f &→ Y &→ C(Y, X) g L15 166 Yoneda functor op y:C → SetC (where C is a small category) is the Curried version of the hom functor Hom C × Cop =∼ Cop × C −→C Set op ! For each C-object X,theobjectyX ∈ SetC is the functor C(_ , X) : Cop → Set given by Z &→ C(Z, X) g ◦ f f &→ f ∗ Y &→ C(Y, X) g this function is often written as f ∗ L15 166 Yoneda functor op y:C → SetC (where C is a small category) is the Curried version of the hom functor Hom C × Cop =∼ Cop × C −→C Set f y f ! For each C-morphism Y −→ X,themorphismyY −→ yX in op SetC is the natural transformation whose component at any given Z ∈ Cop is the function (y f ) yY(Z) Z yX(Z) C(Z, Y) C(Z, X) g f ◦ g L15 166 Yoneda functor op y:C → SetC (where C is a small category) is the Curried version of the hom functor Hom C × Cop =∼ Cop × C −→C Set f y f ! For each C-morphism Y −→ X,themorphismyY −→ yX in op SetC is the natural transformation whose component at any given Z ∈ Cop is the function (y f ) yY(Z) Z yX(Z) this function is often C(Z, Y) C(Z, X) written as f∗ f∗ g f ◦ g L15 166 The Yoneda Lemma For each small category C,eachobjectX ∈ C and each op presheaf F ∈ SetC ,thereisabijectionofsets Cop ∼ ηX,F : Set (yX, F) = F(X) which is natural in both X and F. L15 167 The Yoneda Lemma For each small category C,eachobjectX ∈ C and each op presheaf F ∈ SetC ,thereisabijectionofsets Cop ∼ ηX,F : Set (yX, F) = F(X) which is natural in both X and F. the value of F : Cop → Set at X the set of natural transformations from the functor yX : Cop → Set to the functor F : Cop → Set L15 167 The Yoneda Lemma For each small category C,eachobjectX ∈ C and each op presheaf F ∈ SetC ,thereisabijectionofsets Cop ∼ ηX,F : Set (yX, F) = F(X) which is natural in both X and F. Cop Definition of the function ηX,F : Set (yX, F) → F(X): op for each θ :yX → F in SetC we have the function θ C(X, X)=yX(X) −→X F(X) and define ηX,F(θ) " θX (idX ) L15 167 The Yoneda Lemma For each small category C,eachobjectX ∈ C and each op presheaf F ∈ SetC ,thereisabijectionofsets Cop ∼ ηX,F : Set (yX, F) = F(X) which is natural in both X and F. −1 Cop Definition of the function ηX,F : F(X) → Set (yX, F): for each x ∈ X, Y ∈ C and f ∈ yX(Y)=C(Y, X), F( f ) we get a F(X) −−→ F(Y) in Set and hence F( f)(x) ∈ F(Y); L15 167 The Yoneda Lemma For each small category C,eachobjectX ∈ C and each op presheaf F ∈ SetC ,thereisabijectionofsets Cop ∼ ηX,F : Set (yX, F) = F(X) which is natural in both X and F. −1 Cop Definition of the function ηX,F : F(X) → Set (yX, F): for each x ∈ X, Y ∈ C and f ∈ yX(Y)=C(Y, X), F( f ) we get a F(X) −−→ F(Y) in Set and hence F( f)(x) ∈ F(Y); −1 Define ηX,F(x) :yX(Y) → F(Y) by Y ! " check this gives a −1 " natural transformation η , (x) ( f) F( f)(x) X F Y −1 ηX,F(x) :yX → F L15 ! " 167 − ◦ 1 Proof of ηX,F ηX,F = idF(X) For any x ∈ F(X) we have −1 " −1 ηX,F η , (x) η , (x) (idX ) by definition of ηX,F X F X F X ! " " ! " −1 F(idX )(x) by definition of ηX,F = idF(X)(x) since F is a functor = x L15 168 −1 ◦ op Proof of ηX,F ηX,F = idSetC (yX,F) θ op f For any yX −→ F in SetC and Y −→ X in C,wehave −1 −1 η (ηX,F(θ)) f " η (θX (idX ))) f by definition of ηX,F X,F Y X,F Y ! " " ! " −1 F( f )(θX (idX )) by definition of ηX,F ∗ =θY ( f (idX )) by naturality of θ ∗ " θY (idX ◦ f ) by definition of f = θY ( f ) naturality of θ θ yX(Y) Y F(Y) f ∗ F( f ) yX(X) F(X) θX L15 169 −1 ◦ op Proof of ηX,F ηX,F = idSetC (yX,F) θ op f For any yX −→ F in SetC and Y −→ X in C,wehave −1 −1 η (ηX,F(θ)) f " η (θX (idX ))) f by definition of ηX,F X,F Y X,F Y ! " " ! " −1 F( f )(θX (idX )) by definition of ηX,F ∗ =θY ( f (idX )) by naturality of θ ∗ " θY (idX ◦ f ) by definition of f = θY ( f ) −1 so ∀θ, Y, η (ηX,F(θ)) = θY X,F Y −!1 " so ∀θ, ηX,F (ηX,F(θ)) = θ − 1 ◦ so ηX,F ηX,F = id. L15 169 The Yoneda Lemma For each small category C,eachobjectX ∈ C and each op presheaf F ∈ SetC ,thereisabijectionofsets Cop ∼ ηX,F : Set (yX, F) = F(X) which is natural in both X and F. L15 170 Proof that ηX,F is natural in F: ϕ op Given F −→ G in SetC ,havetoshowthat op ηX,F SetC (yX, F) F(X) ϕ∗ ϕX op SetC (yX, G) G(X) ηX,G θ commutes in Set.ForallyX −→ F we have ϕX (ηX,F(θ)) " ϕX (θX (idX )) " (ϕ ◦ θ)X (idX ) " ηX,G(ϕ ◦ θ) " ηX,G(ϕ∗(θ)) L15 171 Proof that ηX,F is natural in X: f Given Y −→ X in C,havetoshowthat op ηX,F SetC (yX, F) F(X) (y f )∗ F( f ) op SetC (yY, F) F(Y) ηY,F θ commutes in Set.ForallyX −→ F we have F( f )((ηX,F(θ)) " F( f )(θX (idX )) ∗ = θY ( f (idX )) by naturality of θ = θY ( f ) = θY ( f∗(idY )) " (θ ◦ y f )Y (idY ) " ηY,F(θ ◦ y f ) ∗ " ηY,F((y f ) (θ)) L15 172 Corollary of the Yoneda Lemma: op the functor y:C → SetC is full and faithful. In general, a functor F : C → D is ! faithful if for all X, Y ∈ C the function C(X, Y) → D(F(X), F(Y)) f &→ F( f ) is injective: ∀ f, f ′ ∈ C(X, Y), F( f )=F( f ′) ⇒ f = f ′ ! full if the above functions are all surjective: ∀g ∈ D(F(X), F(Y)), ∃ f ∈ C(X, Y), F( f )=g L15 173.