mathlib documentation

category_theory.products.basic

Cartesian products of categories #

We define the category instance on C × D when C and D are categories.

We define:

We further define evaluation : C ⥤ (C ⥤ D) ⥤ D and evaluation_uncurried : C × (C ⥤ D) ⥤ D, and products of functors and natural transformations, written F.prod G and α.prod β.

@[simp]
theorem category_theory.prod_id_fst (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C × D) :
(𝟙 X).fst = 𝟙 X.fst
@[simp]
theorem category_theory.prod_id_snd (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C × D) :
(𝟙 X).snd = 𝟙 X.snd
@[simp]
theorem category_theory.prod_comp_fst (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (_x _x_1 _x_2 : C × D) (f : _x _x_1) (g : _x_1 _x_2) :
(f g).fst = f.fst g.fst
@[simp]
theorem category_theory.prod_hom (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X Y : C × D) :
(X Y) = ((X.fst Y.fst) × (X.snd Y.snd))
@[protected, instance]

prod C D gives the cartesian product of two categories.

See https://stacks.math.columbia.edu/tag/001K.

Equations
@[simp]
theorem category_theory.prod_comp_snd (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (_x _x_1 _x_2 : C × D) (f : _x _x_1) (g : _x_1 _x_2) :
(f g).snd = f.snd g.snd
@[simp]
theorem category_theory.prod_id (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C) (Y : D) :
𝟙 (X, Y) = (𝟙 X, 𝟙 Y)

Two rfl lemmas that cannot be generated by @[simps].

@[simp]
theorem category_theory.prod_comp (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] {P Q R : C} {S T U : D} (f : (P, S) (Q, T)) (g : (Q, T) (R, U)) :
f g = (f.fst g.fst, f.snd g.snd)
def category_theory.prod.eta_iso {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (X : C × D) :
(X.fst, X.snd) X

The isomorphism between (X.1, X.2) and X.

Equations
def category_theory.iso.prod {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {P Q : C} {S T : D} (f : P Q) (g : S T) :
(P, S) (Q, T)

Construct an isomorphism in C × D out of two isomorphisms in C and D.

Equations
@[simp]
theorem category_theory.iso.prod_hom {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {P Q : C} {S T : D} (f : P Q) (g : S T) :
(f.prod g).hom = (f.hom, g.hom)
@[simp]
theorem category_theory.iso.prod_inv {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {P Q : C} {S T : D} (f : P Q) (g : S T) :
(f.prod g).inv = (f.inv, g.inv)
@[protected, instance]

prod.category.uniform C D is an additional instance specialised so both factors have the same universe levels. This helps typeclass resolution.

Equations
@[simp]
theorem category_theory.prod.sectl_map (C : Type u₁) [category_theory.category C] {D : Type u₂} [category_theory.category D] (Z : D) (X Y : C) (f : X Y) :
def category_theory.prod.sectl (C : Type u₁) [category_theory.category C] {D : Type u₂} [category_theory.category D] (Z : D) :
C C × D

sectl C Z is the functor C ⥤ C × D given by X ↦ (X, Z).

Equations
@[simp]
theorem category_theory.prod.sectl_obj (C : Type u₁) [category_theory.category C] {D : Type u₂} [category_theory.category D] (Z : D) (X : C) :
@[simp]
theorem category_theory.prod.sectr_map {C : Type u₁} [category_theory.category C] (Z : C) (D : Type u₂) [category_theory.category D] (X Y : D) (f : X Y) :
def category_theory.prod.sectr {C : Type u₁} [category_theory.category C] (Z : C) (D : Type u₂) [category_theory.category D] :
D C × D

sectr Z D is the functor D ⥤ C × D given by Y ↦ (Z, Y) .

Equations
@[simp]
theorem category_theory.prod.sectr_obj {C : Type u₁} [category_theory.category C] (Z : C) (D : Type u₂) [category_theory.category D] (X : D) :

fst is the functor (X, Y) ↦ X.

Equations
@[simp]
theorem category_theory.prod.fst_map (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X Y : C × D) (f : X Y) :
@[simp]
theorem category_theory.prod.snd_map (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X Y : C × D) (f : X Y) :

snd is the functor (X, Y) ↦ Y.

Equations
@[simp]
@[simp]
theorem category_theory.prod.swap_map (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (_x _x_1 : C × D) (f : _x _x_1) :

The functor swapping the factors of a cartesian product of categories, C × D ⥤ D × C.

Equations
Instances for category_theory.prod.swap

Swapping the factors of a cartesion product of categories twice is naturally isomorphic to the identity functor.

Equations
@[simp]
theorem category_theory.prod.braiding_functor_map (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (_x _x_1 : C × D) (f : _x _x_1) :
@[simp]
theorem category_theory.prod.braiding_inverse_map (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (_x _x_1 : D × C) (f : _x _x_1) :
@[simp]
theorem category_theory.evaluation_map_app (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X Y : C) (f : X Y) (F : C D) :

The "evaluation at X" functor, such that (evaluation.obj X).obj F = F.obj X, which is functorial in both X and F.

Equations
@[simp]
theorem category_theory.evaluation_obj_obj (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C) (F : C D) :
@[simp]
theorem category_theory.evaluation_obj_map (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C) (F G : C D) (α : F G) :

The "evaluation of F at X" functor, as a functor C × (C ⥤ D) ⥤ D.

Equations
@[simp]
theorem category_theory.functor.prod_map {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] {D : Type u₄} [category_theory.category D] (F : A B) (G : C D) (_x _x_1 : A × C) (f : _x _x_1) :
(F.prod G).map f = (F.map f.fst, G.map f.snd)
@[simp]
theorem category_theory.functor.prod_obj {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] {D : Type u₄} [category_theory.category D] (F : A B) (G : C D) (X : A × C) :
(F.prod G).obj X = (F.obj X.fst, G.obj X.snd)
def category_theory.functor.prod {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] {D : Type u₄} [category_theory.category D] (F : A B) (G : C D) :
A × C B × D

The cartesian product of two functors.

Equations
@[simp]
theorem category_theory.functor.prod'_map {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A B) (G : A C) (x y : A) (f : x y) :
(F.prod' G).map f = (F.map f, G.map f)
@[simp]
theorem category_theory.functor.prod'_obj {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A B) (G : A C) (a : A) :
(F.prod' G).obj a = (F.obj a, G.obj a)
def category_theory.functor.prod' {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A B) (G : A C) :
A B × C

Similar to prod, but both functors start from the same category A

Equations
@[simp]
theorem category_theory.functor.prod'_comp_fst_inv_app {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A B) (G : A C) (X : A) :
(F.prod'_comp_fst G).inv.app X = 𝟙 (F.obj X)

The product F.prod' G followed by projection on the first component is isomorphic to F

Equations
@[simp]
theorem category_theory.functor.prod'_comp_fst_hom_app {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A B) (G : A C) (X : A) :
(F.prod'_comp_fst G).hom.app X = 𝟙 (F.obj X)
@[simp]
theorem category_theory.functor.prod'_comp_snd_hom_app {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A B) (G : A C) (X : A) :
(F.prod'_comp_snd G).hom.app X = 𝟙 (G.obj X)

The product F.prod' G followed by projection on the second component is isomorphic to G

Equations
@[simp]
theorem category_theory.functor.prod'_comp_snd_inv_app {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A B) (G : A C) (X : A) :
(F.prod'_comp_snd G).inv.app X = 𝟙 (G.obj X)

The diagonal functor.

Equations
@[simp]
theorem category_theory.functor.diag_map (C : Type u₃) [category_theory.category C] {X Y : C} (f : X Y) :
def category_theory.nat_trans.prod {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] {D : Type u₄} [category_theory.category D] {F G : A B} {H I : C D} (α : F G) (β : H I) :
F.prod H G.prod I

The cartesian product of two natural transformations.

Equations
@[simp]
theorem category_theory.nat_trans.prod_app {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] {D : Type u₄} [category_theory.category D] {F G : A B} {H I : C D} (α : F G) (β : H I) (X : A × C) :

F.flip composed with evaluation is the same as evaluating F.

Equations
@[simp]

The forward direction for functor_prod_functor_equiv

Equations

The backward direction for functor_prod_functor_equiv

Equations
@[simp]
theorem category_theory.functor_prod_to_prod_functor_map (A : Type u₁) [category_theory.category A] (B : Type u₂) [category_theory.category B] (C : Type u₃) [category_theory.category C] (F G : A B × C) (α : F G) :
(category_theory.functor_prod_to_prod_functor A B C).map α = ({app := λ (X : A), (α.app X).fst, naturality' := _}, {app := λ (X : A), (α.app X).snd, naturality' := _})