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Mathlib.CategoryTheory.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 evaluationUncurried : C × (C ⥤ D) ⥤ D, and products of functors and natural transformations, written F.prod G and α.prod β.

@[simp]
theorem CategoryTheory.prod_Hom (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] (X : C × D) (Y : C × D) :
(X Y) = ((X.1 Y.1) × (X.2 Y.2))
@[simp]

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

@[simp]
theorem CategoryTheory.prod_comp (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] {P : C} {Q : C} {R : C} {S : D} {T : D} {U : D} (f : (P, S) (Q, T)) (g : (Q, T) (R, U)) :

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

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    @[simp]
    theorem CategoryTheory.Iso.prod_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {P : C} {Q : C} {S : D} {T : D} (f : P Q) (g : S T) :
    (f.prod g).hom = (f.hom, g.hom)
    @[simp]
    theorem CategoryTheory.Iso.prod_inv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {P : C} {Q : C} {S : D} {T : D} (f : P Q) (g : S T) :
    (f.prod g).inv = (f.inv, g.inv)
    def CategoryTheory.Iso.prod {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {P : C} {Q : C} {S : D} {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.

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    • f.prod g = { hom := (f.hom, g.hom), inv := (f.inv, g.inv), hom_inv_id := , inv_hom_id := }
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      Category.uniformProd C D is an additional instance specialised so both factors have the same universe levels. This helps typeclass resolution.

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

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

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          fst is the functor (X, Y) ↦ X.

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            snd is the functor (X, Y) ↦ Y.

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              theorem CategoryTheory.Prod.swap_map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] :
              ∀ {X Y : C × D} (f : X Y), (CategoryTheory.Prod.swap C D).map f = (f.2, f.1)

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

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

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                  @[simp]
                  theorem CategoryTheory.Prod.braiding_inverse_map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] :
                  ∀ {X Y : D × C} (f : X Y), (CategoryTheory.Prod.braiding C D).inverse.map f = (f.2, f.1)
                  @[simp]
                  theorem CategoryTheory.Prod.braiding_functor_map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] :
                  ∀ {X Y : C × D} (f : X Y), (CategoryTheory.Prod.braiding C D).functor.map f = (f.2, f.1)

                  The equivalence, given by swapping factors, between C × D and D × C.

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                    @[simp]
                    theorem CategoryTheory.evaluation_obj_map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] (X : C) :
                    ∀ {X_1 Y : CategoryTheory.Functor C D} (α : X_1 Y), ((CategoryTheory.evaluation C D).obj X).map α = α.app X
                    @[simp]
                    theorem CategoryTheory.evaluation_map_app (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] {X : C} {Y : C} (f : X Y) (F : CategoryTheory.Functor C D) :
                    ((CategoryTheory.evaluation C D).map f).app F = F.map f

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

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

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                        The constant functor followed by the evaluation functor is just the identity.

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                          @[simp]
                          theorem CategoryTheory.Functor.prod_map {A : Type u₁} [CategoryTheory.Category.{v₁, u₁} A] {B : Type u₂} [CategoryTheory.Category.{v₂, u₂} B] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄} [CategoryTheory.Category.{v₄, u₄} D] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C D) :
                          ∀ {X Y : A × C} (f : X Y), (F.prod G).map f = (F.map f.1, G.map f.2)

                          The cartesian product of two functors.

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                          • F.prod G = { obj := fun (X : A × C) => (F.obj X.1, G.obj X.2), map := fun {X Y : A × C} (f : X Y) => (F.map f.1, G.map f.2), map_id := , map_comp := }
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                            @[simp]
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                            theorem CategoryTheory.Functor.prod'_map {A : Type u₁} [CategoryTheory.Category.{v₁, u₁} A] {B : Type u₂} [CategoryTheory.Category.{v₂, u₂} B] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor A C) :
                            ∀ {X Y : A} (f : X Y), (F.prod' G).map f = (F.map f, G.map f)

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

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                            • F.prod' G = { obj := fun (a : A) => (F.obj a, G.obj a), map := fun {X Y : A} (f : X Y) => (F.map f, G.map f), map_id := , map_comp := }
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                              The product F.prod' G followed by projection on the first component is isomorphic to F

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

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                                  @[simp]
                                  theorem CategoryTheory.Functor.diag_map (C : Type u₃) [CategoryTheory.Category.{v₃, u₃} C] {X : C} {Y : C} (f : X Y) :

                                  The cartesian product of two natural transformations.

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                                    The cartesian product of two natural isomorphisms.

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                                      @[simp]
                                      theorem CategoryTheory.Equivalence.prod_functor {A : Type u₁} [CategoryTheory.Category.{v₁, u₁} A] {B : Type u₂} [CategoryTheory.Category.{v₂, u₂} B] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄} [CategoryTheory.Category.{v₄, u₄} D] (E₁ : A B) (E₂ : C D) :
                                      (E₁.prod E₂).functor = E₁.functor.prod E₂.functor
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                                      theorem CategoryTheory.Equivalence.prod_inverse {A : Type u₁} [CategoryTheory.Category.{v₁, u₁} A] {B : Type u₂} [CategoryTheory.Category.{v₂, u₂} B] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄} [CategoryTheory.Category.{v₄, u₄} D] (E₁ : A B) (E₂ : C D) :
                                      (E₁.prod E₂).inverse = E₁.inverse.prod E₂.inverse
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                                      theorem CategoryTheory.Equivalence.prod_counitIso {A : Type u₁} [CategoryTheory.Category.{v₁, u₁} A] {B : Type u₂} [CategoryTheory.Category.{v₂, u₂} B] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄} [CategoryTheory.Category.{v₄, u₄} D] (E₁ : A B) (E₂ : C D) :
                                      (E₁.prod E₂).counitIso = CategoryTheory.NatIso.prod E₁.counitIso E₂.counitIso
                                      @[simp]
                                      theorem CategoryTheory.Equivalence.prod_unitIso {A : Type u₁} [CategoryTheory.Category.{v₁, u₁} A] {B : Type u₂} [CategoryTheory.Category.{v₂, u₂} B] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄} [CategoryTheory.Category.{v₄, u₄} D] (E₁ : A B) (E₂ : C D) :
                                      (E₁.prod E₂).unitIso = CategoryTheory.NatIso.prod E₁.unitIso E₂.unitIso

                                      The cartesian product of two equivalences of categories.

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

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                                          The forward direction for functorProdFunctorEquiv

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                                            @[simp]
                                            theorem CategoryTheory.functorProdToProdFunctor_map (A : Type u₁) [CategoryTheory.Category.{v₁, u₁} A] (B : Type u₂) [CategoryTheory.Category.{v₂, u₂} B] (C : Type u₃) [CategoryTheory.Category.{v₃, u₃} C] :
                                            ∀ {X Y : CategoryTheory.Functor A (B × C)} (α : X Y), (CategoryTheory.functorProdToProdFunctor A B C).map α = ({ app := fun (X_1 : A) => (α.app X_1).1, naturality := }, { app := fun (X_1 : A) => (α.app X_1).2, naturality := })

                                            The backward direction for functorProdFunctorEquiv

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                                              The equivalence of categories between (A ⥤ B) × (A ⥤ C) and A ⥤ (B × C)

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                                                theorem CategoryTheory.prodOpEquiv_inverse_map (C : Type u₃) [CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄} [CategoryTheory.Category.{v₄, u₄} D] :
                                                ∀ {X Y : Cᵒᵖ × Dᵒᵖ} (x : X Y), (CategoryTheory.prodOpEquiv C).inverse.map x = match x with | (f, g) => Opposite.op (f.unop, g.unop)
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                                                theorem CategoryTheory.prodOpEquiv_functor_map (C : Type u₃) [CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄} [CategoryTheory.Category.{v₄, u₄} D] :
                                                ∀ {X Y : (C × D)ᵒᵖ} (f : X Y), (CategoryTheory.prodOpEquiv C).functor.map f = (f.unop.1.op, f.unop.2.op)
                                                @[simp]
                                                theorem CategoryTheory.prodOpEquiv_counitIso (C : Type u₃) [CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄} [CategoryTheory.Category.{v₄, u₄} D] :
                                                (CategoryTheory.prodOpEquiv C).counitIso = CategoryTheory.Iso.refl ({ obj := fun (x : Cᵒᵖ × Dᵒᵖ) => match x with | (X, Y) => Opposite.op (Opposite.unop X, Opposite.unop Y), map := fun {X Y : Cᵒᵖ × Dᵒᵖ} (x : X Y) => match x with | (f, g) => Opposite.op (f.unop, g.unop), map_id := , map_comp := }.comp { obj := fun (X : (C × D)ᵒᵖ) => (Opposite.op (Opposite.unop X).1, Opposite.op (Opposite.unop X).2), map := fun {X Y : (C × D)ᵒᵖ} (f : X Y) => (f.unop.1.op, f.unop.2.op), map_id := , map_comp := })

                                                The equivalence between the opposite of a product and the product of the opposites.

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