Documentation

Mathlib.CategoryTheory.Preadditive.AdditiveFunctor

Additive Functors #

A functor between two preadditive categories is called additive provided that the induced map on hom types is a morphism of abelian groups.

An additive functor between preadditive categories creates and preserves biproducts. Conversely, if F : C ⥤ D is a functor between preadditive categories, where C has binary biproducts, and if F preserves binary biproducts, then F is additive.

We also define the category of bundled additive functors.

Implementation details #

Functor.Additive is a Prop-valued class, defined by saying that for every two objects X and Y, the map F.map : (X ⟶ Y) → (F.obj X ⟶ F.obj Y) is a morphism of abelian groups.

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class CategoryTheory.Functor.Additive {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Preadditive C] [Preadditive D] (F : Functor C D) :
Prop

A functor F is additive provided F.map is an additive homomorphism.

  • map_add {X Y : C} {f g : X ⟶ Y} : F.map (f + g) = F.map f + F.map g

    the addition of two morphisms is mapped to the sum of their images

Instances
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@[simp]
theorem CategoryTheory.Functor.map_add {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_5, u_2} D] [Preadditive C] [Preadditive D] (F : Functor C D) [F.Additive] {X Y : C} {f g : X ⟶ Y} :
F.map (f + g) = F.map f + F.map g
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def CategoryTheory.Functor.mapAddHom {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_5, u_2} D] [Preadditive C] [Preadditive D] (F : Functor C D) [F.Additive] {X Y : C} :
(X ⟶ Y) →+ (F.obj X ⟶ F.obj Y)

F.mapAddHom is an additive homomorphism whose underlying function is F.map.

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@[simp]
theorem CategoryTheory.Functor.mapAddHom_apply {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_5, u_2} D] [Preadditive C] [Preadditive D] (F : Functor C D) [F.Additive] {X Y : C} (f : X ⟶ Y) :
F.mapAddHom f = F.map f
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theorem CategoryTheory.Functor.coe_mapAddHom {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_5, u_2} D] [Preadditive C] [Preadditive D] (F : Functor C D) [F.Additive] {X Y : C} :
⇑F.mapAddHom = F.map
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@[instance 100]
instance CategoryTheory.Functor.preservesZeroMorphisms_of_additive {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_5, u_2} D] [Preadditive C] [Preadditive D] (F : Functor C D) [F.Additive] :
F.PreservesZeroMorphisms
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instance CategoryTheory.Functor.instAdditiveId {C : Type u_1} [Category.{u_4, u_1} C] [Preadditive C] :
(Functor.id C).Additive
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instance CategoryTheory.Functor.instAdditiveComp {C : Type u_1} {D : Type u_2} [Category.{u_7, u_1} C] [Category.{u_6, u_2} D] [Preadditive C] [Preadditive D] (F : Functor C D) [F.Additive] {E : Type u_4} [Category.{u_5, u_4} E] [Preadditive E] (G : Functor D E) [G.Additive] :
(F.comp G).Additive
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instance CategoryTheory.Functor.instAdditiveObjEvaluation {C : Type u_1} [Category.{u_6, u_1} C] [Preadditive C] {J : Type u_4} [Category.{u_5, u_4} J] (j : J) :
((evaluation J C).obj j).Additive
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@[simp]
theorem CategoryTheory.Functor.map_neg {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_5, u_2} D] [Preadditive C] [Preadditive D] (F : Functor C D) [F.Additive] {X Y : C} {f : X ⟶ Y} :
F.map (-f) = -F.map f
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@[simp]
theorem CategoryTheory.Functor.map_sub {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_5, u_2} D] [Preadditive C] [Preadditive D] (F : Functor C D) [F.Additive] {X Y : C} {f g : X ⟶ Y} :
F.map (f - g) = F.map f - F.map g
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theorem CategoryTheory.Functor.map_nsmul {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_5, u_2} D] [Preadditive C] [Preadditive D] (F : Functor C D) [F.Additive] {X Y : C} {f : X ⟶ Y} {n : ℕ} :
F.map (n • f) = n • F.map f
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theorem CategoryTheory.Functor.map_zsmul {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_5, u_2} D] [Preadditive C] [Preadditive D] (F : Functor C D) [F.Additive] {X Y : C} {f : X ⟶ Y} {r : ℤ} :
F.map (r • f) = r • F.map f
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@[simp]
theorem CategoryTheory.Functor.map_sum {C : Type u_1} {D : Type u_2} [Category.{u_5, u_1} C] [Category.{u_6, u_2} D] [Preadditive C] [Preadditive D] (F : Functor C D) [F.Additive] {X Y : C} {α : Type u_4} (f : α → (X ⟶ Y)) (s : Finset α) :
F.map (∑ a ∈ s, f a) = ∑ a ∈ s, F.map (f a)
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theorem CategoryTheory.Functor.additive_of_iso {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_5, u_2} D] [Preadditive C] [Preadditive D] {F : Functor C D} [F.Additive] {G : Functor C D} (e : F ≅ G) :
G.Additive
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theorem CategoryTheory.Functor.additive_of_full_essSurj_comp {C : Type u_1} {D : Type u_2} {E : Type u_3} [Category.{u_4, u_1} C] [Category.{u_5, u_2} D] [Category.{u_6, u_3} E] [Preadditive C] [Preadditive D] [Preadditive E] (F : Functor C D) [F.Additive] [F.Full] [F.EssSurj] (G : Functor D E) [(F.comp G).Additive] :
G.Additive
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theorem CategoryTheory.Functor.additive_of_comp_faithful {C : Type u_1} {D : Type u_2} {E : Type u_3} [Category.{u_4, u_1} C] [Category.{u_5, u_2} D] [Category.{u_6, u_3} E] [Preadditive C] [Preadditive D] [Preadditive E] (F : Functor C D) (G : Functor D E) [G.Additive] [(F.comp G).Additive] [G.Faithful] :
F.Additive
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theorem CategoryTheory.Functor.hasZeroObject_of_additive {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_5, u_2} D] [Preadditive C] [Preadditive D] (F : Functor C D) [F.Additive] [Limits.HasZeroObject C] :
Limits.HasZeroObject D
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instance CategoryTheory.Functor.inducedFunctor_additive {C : Type u_1} {D : Type u_2} [Category.{u_3, u_2} D] [Preadditive D] (F : C → D) :
(inducedFunctor F).Additive
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instance CategoryTheory.Functor.fullSubcategoryInclusion_additive {C : Type u_1} [Category.{u_2, u_1} C] [Preadditive C] (Z : ObjectProperty C) :
Z.ι.Additive
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@[instance 100]
instance CategoryTheory.Functor.preservesFiniteBiproductsOfAdditive {C : Type u₁} {D : Type u₂} [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] (F : Functor C D) [F.Additive] :
Limits.PreservesFiniteBiproducts F
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@[instance 100]
instance CategoryTheory.Functor.preservesFiniteCoproductsOfAdditive {C : Type u₁} {D : Type u₂} [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] (F : Functor C D) [F.Additive] :
Limits.PreservesFiniteCoproducts F
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@[instance 100]
instance CategoryTheory.Functor.preservesFiniteProductsOfAdditive {C : Type u₁} {D : Type u₂} [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] (F : Functor C D) [F.Additive] :
Limits.PreservesFiniteProducts F
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theorem CategoryTheory.Functor.additive_of_preservesBinaryBiproducts {C : Type u₁} {D : Type u₂} [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] (F : Functor C D) [Limits.HasBinaryBiproducts C] [F.PreservesZeroMorphisms] [Limits.PreservesBinaryBiproducts F] :
F.Additive
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theorem CategoryTheory.Functor.additive_of_preserves_binary_products {C : Type u₁} {D : Type u₂} [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] (F : Functor C D) [Limits.HasBinaryProducts C] [Limits.PreservesLimitsOfShape (Discrete Limits.WalkingPair) F] [F.PreservesZeroMorphisms] :
F.Additive
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instance CategoryTheory.Equivalence.inverse_additive {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Preadditive C] [Preadditive D] (e : C ≌ D) [e.functor.Additive] :
e.inverse.Additive
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def CategoryTheory.AdditiveFunctor (C : Type u_1) (D : Type u_2) [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Preadditive C] [Preadditive D] :
Type (max (max (max u_1 u_2) u_3) u_4)

Bundled additive functors.

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instance CategoryTheory.instCategoryAdditiveFunctor (C : Type u_1) (D : Type u_2) [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Preadditive C] [Preadditive D] :
Category.{max u_1 u_4, max (max (max u_4 u_3) u_2) u_1} (C ⥤+ D)
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def CategoryTheory.«term_⥤+_» :
Lean.TrailingParserDescr

the category of additive functors is denoted C ⥤+ D

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instance CategoryTheory.instPreadditiveAdditiveFunctor (C : Type u_1) (D : Type u_2) [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Preadditive C] [Preadditive D] :
Preadditive (C ⥤+ D)
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def CategoryTheory.AdditiveFunctor.forget (C : Type u_1) (D : Type u_2) [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Preadditive C] [Preadditive D] :
Functor (C ⥤+ D) (Functor C D)

An additive functor is in particular a functor.

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instance CategoryTheory.instFullAdditiveFunctorFunctorForget (C : Type u_1) (D : Type u_2) [Category.{u_4, u_1} C] [Category.{u_3, u_2} D] [Preadditive C] [Preadditive D] :
(AdditiveFunctor.forget C D).Full
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instance CategoryTheory.instFaithfulAdditiveFunctorFunctorForget (C : Type u_1) (D : Type u_2) [Category.{u_4, u_1} C] [Category.{u_3, u_2} D] [Preadditive C] [Preadditive D] :
(AdditiveFunctor.forget C D).Faithful
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def CategoryTheory.AdditiveFunctor.of {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Preadditive C] [Preadditive D] (F : Functor C D) [F.Additive] :
C ⥤+ D

Turn an additive functor into an object of the category AdditiveFunctor C D.

Equations
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@[simp]
theorem CategoryTheory.AdditiveFunctor.of_fst {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Preadditive C] [Preadditive D] (F : Functor C D) [F.Additive] :
(of F).obj = F
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@[simp]
theorem CategoryTheory.AdditiveFunctor.forget_obj {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Preadditive C] [Preadditive D] (F : C ⥤+ D) :
(forget C D).obj F = F.obj
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theorem CategoryTheory.AdditiveFunctor.forget_obj_of {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Preadditive C] [Preadditive D] (F : Functor C D) [F.Additive] :
(forget C D).obj (of F) = F
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@[simp]
theorem CategoryTheory.AdditiveFunctor.forget_map {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Preadditive C] [Preadditive D] (F G : C ⥤+ D) (α : F ⟶ G) :
(forget C D).map α = α
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instance CategoryTheory.instAdditiveAdditiveFunctorFunctorForget {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Preadditive C] [Preadditive D] :
(AdditiveFunctor.forget C D).Additive
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instance CategoryTheory.instAdditiveObjFunctor {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] [Preadditive C] [Preadditive D] (F : C ⥤+ D) :
F.obj.Additive
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def CategoryTheory.AdditiveFunctor.ofLeftExact (C : Type u₁) (D : Type u₂) [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] [Limits.HasZeroObject C] [Limits.HasZeroObject D] [Limits.HasBinaryBiproducts C] :
Functor (C ⥤ₗ D) (C ⥤+ D)

Turn a left exact functor into an additive functor.

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instance CategoryTheory.instFullLeftExactFunctorAdditiveFunctorOfLeftExact (C : Type u₁) (D : Type u₂) [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] [Limits.HasZeroObject C] [Limits.HasZeroObject D] [Limits.HasBinaryBiproducts C] :
(AdditiveFunctor.ofLeftExact C D).Full
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instance CategoryTheory.instFaithfulLeftExactFunctorAdditiveFunctorOfLeftExact (C : Type u₁) (D : Type u₂) [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] [Limits.HasZeroObject C] [Limits.HasZeroObject D] [Limits.HasBinaryBiproducts C] :
(AdditiveFunctor.ofLeftExact C D).Faithful
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def CategoryTheory.AdditiveFunctor.ofRightExact (C : Type u₁) (D : Type u₂) [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] [Limits.HasZeroObject C] [Limits.HasZeroObject D] [Limits.HasBinaryBiproducts C] :
Functor (C ⥤ᵣ D) (C ⥤+ D)

Turn a right exact functor into an additive functor.

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instance CategoryTheory.instFullRightExactFunctorAdditiveFunctorOfRightExact (C : Type u₁) (D : Type u₂) [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] [Limits.HasZeroObject C] [Limits.HasZeroObject D] [Limits.HasBinaryBiproducts C] :
(AdditiveFunctor.ofRightExact C D).Full
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instance CategoryTheory.instFaithfulRightExactFunctorAdditiveFunctorOfRightExact (C : Type u₁) (D : Type u₂) [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] [Limits.HasZeroObject C] [Limits.HasZeroObject D] [Limits.HasBinaryBiproducts C] :
(AdditiveFunctor.ofRightExact C D).Faithful
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def CategoryTheory.AdditiveFunctor.ofExact (C : Type u₁) (D : Type u₂) [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] [Limits.HasZeroObject C] [Limits.HasZeroObject D] [Limits.HasBinaryBiproducts C] :
Functor (C ⥤ₑ D) (C ⥤+ D)

Turn an exact functor into an additive functor.

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instance CategoryTheory.instFullExactFunctorAdditiveFunctorOfExact (C : Type u₁) (D : Type u₂) [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] [Limits.HasZeroObject C] [Limits.HasZeroObject D] [Limits.HasBinaryBiproducts C] :
(AdditiveFunctor.ofExact C D).Full
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instance CategoryTheory.instFaithfulExactFunctorAdditiveFunctorOfExact (C : Type u₁) (D : Type u₂) [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] [Limits.HasZeroObject C] [Limits.HasZeroObject D] [Limits.HasBinaryBiproducts C] :
(AdditiveFunctor.ofExact C D).Faithful
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@[simp]
theorem CategoryTheory.AdditiveFunctor.ofLeftExact_obj_fst {C : Type u₁} {D : Type u₂} [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] [Limits.HasZeroObject C] [Limits.HasZeroObject D] [Limits.HasBinaryBiproducts C] (F : C ⥤ₗ D) :
((ofLeftExact C D).obj F).obj = F.obj
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@[simp]
theorem CategoryTheory.AdditiveFunctor.ofRightExact_obj_fst {C : Type u₁} {D : Type u₂} [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] [Limits.HasZeroObject C] [Limits.HasZeroObject D] [Limits.HasBinaryBiproducts C] (F : C ⥤ᵣ D) :
((ofRightExact C D).obj F).obj = F.obj
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@[simp]
theorem CategoryTheory.AdditiveFunctor.ofExact_obj_fst {C : Type u₁} {D : Type u₂} [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] [Limits.HasZeroObject C] [Limits.HasZeroObject D] [Limits.HasBinaryBiproducts C] (F : C ⥤ₑ D) :
((ofExact C D).obj F).obj = F.obj
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@[simp]
theorem CategoryTheory.AdditiveFunctor.ofLeftExact_map {C : Type u₁} {D : Type u₂} [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] [Limits.HasZeroObject C] [Limits.HasZeroObject D] [Limits.HasBinaryBiproducts C] {F G : C ⥤ₗ D} (α : F ⟶ G) :
(ofLeftExact C D).map α = α
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@[simp]
theorem CategoryTheory.AdditiveFunctor.ofRightExact_map {C : Type u₁} {D : Type u₂} [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] [Limits.HasZeroObject C] [Limits.HasZeroObject D] [Limits.HasBinaryBiproducts C] {F G : C ⥤ᵣ D} (α : F ⟶ G) :
(ofRightExact C D).map α = α
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@[simp]
theorem CategoryTheory.AdditiveFunctor.ofExact_map {C : Type u₁} {D : Type u₂} [Category.{v₁, u₁} C] [Category.{v₂, u₂} D] [Preadditive C] [Preadditive D] [Limits.HasZeroObject C] [Limits.HasZeroObject D] [Limits.HasBinaryBiproducts C] {F G : C ⥤ₑ D} (α : F ⟶ G) :
(ofExact C D).map α = α