mathlib documentation

category_theory.preadditive.opposite

If C is preadditive, Cᵒᵖ has a natural preadditive structure. #

source
@[protected, instance]
def category_theory.opposite.preadditive (C : Type u_1) [category_theory.category C] [category_theory.preadditive C] :
category_theory.preadditive Cᵒᵖ
Equations
source
@[protected, instance]
def category_theory.module_End_left (C : Type u_1) [category_theory.category C] [category_theory.preadditive C] {X : Cᵒᵖ} {Y : C} :
module (category_theory.End X) (opposite.unop X ⟶ Y)
Equations
source
@[simp]
theorem category_theory.unop_zero (C : Type u_1) [category_theory.category C] [category_theory.preadditive C] (X Y : Cᵒᵖ) :
0.unop = 0
source
@[simp]
theorem category_theory.unop_add (C : Type u_1) [category_theory.category C] [category_theory.preadditive C] {X Y : Cᵒᵖ} (f g : X ⟶ Y) :
(f + g).unop = f.unop + g.unop
source
@[simp]
theorem category_theory.op_zero (C : Type u_1) [category_theory.category C] [category_theory.preadditive C] (X Y : C) :
0.op = 0
source
@[simp]
theorem category_theory.op_add (C : Type u_1) [category_theory.category C] [category_theory.preadditive C] {X Y : C} (f g : X ⟶ Y) :
(f + g).op = f.op + g.op
source
@[protected, instance]
def category_theory.functor.op_additive {C : Type u_1} [category_theory.category C] [category_theory.preadditive C] {D : Type u_3} [category_theory.category D] [category_theory.preadditive D] (F : C ⥤ D) [F.additive] :
F.op.additive
source
@[protected, instance]
def category_theory.functor.right_op_additive {C : Type u_1} [category_theory.category C] [category_theory.preadditive C] {D : Type u_3} [category_theory.category D] [category_theory.preadditive D] (F : Cᵒᵖ ⥤ D) [F.additive] :
F.right_op.additive
source
@[protected, instance]
def category_theory.functor.left_op_additive {C : Type u_1} [category_theory.category C] [category_theory.preadditive C] {D : Type u_3} [category_theory.category D] [category_theory.preadditive D] (F : C ⥤ Dᵒᵖ) [F.additive] :
F.left_op.additive
source
@[protected, instance]
def category_theory.functor.unop_additive {C : Type u_1} [category_theory.category C] [category_theory.preadditive C] {D : Type u_3} [category_theory.category D] [category_theory.preadditive D] (F : Cᵒᵖ ⥤ Dᵒᵖ) [F.additive] :
F.unop.additive