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.

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

@[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

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.

Equations

• F.mapAddHom = AddMonoidHom.mk' (fun (f : X ⟶ Y) => F.map f) ⋯

@[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

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

@[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

instance CategoryTheory.Functor.instAdditiveId {C : Type u_1} [Category.{u_4, u_1} C] [Preadditive C] : (Functor.id C).Additive

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

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

@[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

@[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

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

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

@[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)

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

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

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

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

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

instance CategoryTheory.Functor.fullSubcategoryInclusion_additive {C : Type u_1} [Category.{u_2, u_1} C] [Preadditive C] (Z : C → Prop) : (fullSubcategoryInclusion Z).Additive

@[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

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

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

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

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.

Equations

• (C ⥤+ D) = CategoryTheory.FullSubcategory fun (F : CategoryTheory.Functor C D) => F.Additive

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)

Equations

• CategoryTheory.instCategoryAdditiveFunctor C D = CategoryTheory.FullSubcategory.category fun (F : CategoryTheory.Functor C D) => F.Additive

def CategoryTheory.«term_⥤+_» : Lean.TrailingParserDescr

the category of additive functors is denoted C ⥤+ D

Equations

• CategoryTheory.«term_⥤+_» = Lean.ParserDescr.trailingNode `CategoryTheory.«term_⥤+_» 26 27 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ⥤+ ") (Lean.ParserDescr.cat `term 26))

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)

Equations

• CategoryTheory.instPreadditiveAdditiveFunctor C D = CategoryTheory.Preadditive.inducedCategory CategoryTheory.FullSubcategory.obj

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.

Equations

• CategoryTheory.AdditiveFunctor.forget C D = CategoryTheory.fullSubcategoryInclusion fun (F : CategoryTheory.Functor C D) => F.Additive

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

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

• CategoryTheory.AdditiveFunctor.of F = { obj := F, property := ⋯ }

@[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

@[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

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

@[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

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

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

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.

Equations

• CategoryTheory.AdditiveFunctor.ofLeftExact C D = CategoryTheory.FullSubcategory.map ⋯

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

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

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.

Equations

• CategoryTheory.AdditiveFunctor.ofRightExact C D = CategoryTheory.FullSubcategory.map ⋯

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

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

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.

Equations

• CategoryTheory.AdditiveFunctor.ofExact C D = CategoryTheory.FullSubcategory.map ⋯

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

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

@[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

@[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

@[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

@[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

@[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

@[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