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category_theory.preadditive.additive_functor

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.

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.

Project: #

@[simp]
@[simp]
theorem category_theory.functor.map_add {C : Type u_1} {D : Type u_2} [category_theory.category C] [category_theory.category D] [category_theory.preadditive C] [category_theory.preadditive D] (F : C D) [F.additive] {X Y : C} {f g : X Y} :
F.map (f + g) = F.map f + F.map g
@[simp]

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

Equations
@[simp]
theorem category_theory.functor.map_neg {C : Type u_1} {D : Type u_2} [category_theory.category C] [category_theory.category D] [category_theory.preadditive C] [category_theory.preadditive D] (F : C D) [F.additive] {X Y : C} {f : X Y} :
F.map (-f) = -F.map f
@[simp]
theorem category_theory.functor.map_sub {C : Type u_1} {D : Type u_2} [category_theory.category C] [category_theory.category D] [category_theory.preadditive C] [category_theory.preadditive D] (F : C D) [F.additive] {X Y : C} {f g : X Y} :
F.map (f - g) = F.map f - F.map g
theorem category_theory.functor.map_gsmul {C : Type u_1} {D : Type u_2} [category_theory.category C] [category_theory.category D] [category_theory.preadditive C] [category_theory.preadditive D] (F : C D) [F.additive] {X Y : C} {f : X Y} {r : } :
F.map (r f) = r F.map f
@[simp]
theorem category_theory.functor.map_sum {C : Type u_1} {D : Type u_2} [category_theory.category C] [category_theory.category D] [category_theory.preadditive C] [category_theory.preadditive D] (F : C D) [F.additive] {X Y : C} {α : Type u_5} (f : α → (X Y)) (s : finset α) :
F.map (∑ (a : α) in s, f a) = ∑ (a : α) in s, F.map (f a)

An additive functor takes the zero object to the zero object (up to isomorphism).

Equations
@[instance]

An additive functor between preadditive categories creates finite biproducts.

def category_theory.functor.map_biproduct {C : Type u₁} {D : Type u₂} [category_theory.category C] [category_theory.category D] [category_theory.preadditive C] [category_theory.preadditive D] (F : C D) [F.additive] {J : Type v} [fintype J] [decidable_eq J] (f : J → C) [category_theory.limits.has_biproduct f] :
F.obj ( f) λ (j : J), F.obj (f j)

An additive functor between preadditive categories preserves finite biproducts.

Equations