Preadditive structure on algebras over a monad

If C is a preadditive category and T is an additive monad on C then Algebra T is also preadditive. Dually, if U is an additive comonad on C then Coalgebra U is preadditive as well.

@[simp]

theorem CategoryTheory.Monad.algebraPreadditive_homGroup_zsmul_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (T : CategoryTheory.Monad C) [T.Additive] (F : T.Algebra) (G : T.Algebra) (r : ℤ) (α : F ⟶ G) : (r • α).f = r • α.f

@[simp]

theorem CategoryTheory.Monad.algebraPreadditive_homGroup_sub_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (T : CategoryTheory.Monad C) [T.Additive] (F : T.Algebra) (G : T.Algebra) (α : F ⟶ G) (β : F ⟶ G) : (α - β).f = α.f - β.f

@[simp]

theorem CategoryTheory.Monad.algebraPreadditive_homGroup_neg_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (T : CategoryTheory.Monad C) [T.Additive] (F : T.Algebra) (G : T.Algebra) (α : F ⟶ G) : (-α).f = -α.f

@[simp]

theorem CategoryTheory.Monad.algebraPreadditive_homGroup_add_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (T : CategoryTheory.Monad C) [T.Additive] (F : T.Algebra) (G : T.Algebra) (α : F ⟶ G) (β : F ⟶ G) : (α + β).f = α.f + β.f

@[simp]

theorem CategoryTheory.Monad.algebraPreadditive_homGroup_zero_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (T : CategoryTheory.Monad C) [T.Additive] (F : T.Algebra) (G : T.Algebra) : CategoryTheory.Monad.Algebra.Hom.f 0 = 0

@[simp]

theorem CategoryTheory.Monad.algebraPreadditive_homGroup_nsmul_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (T : CategoryTheory.Monad C) [T.Additive] (F : T.Algebra) (G : T.Algebra) (n : ℕ) (α : F ⟶ G) : (n • α).f = n • α.f

instance CategoryTheory.Monad.algebraPreadditive (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (T : CategoryTheory.Monad C) [T.Additive] : CategoryTheory.Preadditive T.Algebra

The category of algebras over an additive monad on a preadditive category is preadditive.

Equations

• CategoryTheory.Monad.algebraPreadditive C T = { homGroup := fun (F G : T.Algebra) => AddCommGroup.mk ⋯, add_comp := ⋯, comp_add := ⋯ }

instance CategoryTheory.Monad.forget_additive (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (T : CategoryTheory.Monad C) [T.Additive] : T.forget.Additive

Equations

• ⋯ = ⋯

@[simp]

theorem CategoryTheory.Comonad.coalgebraPreadditive_homGroup_neg_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (U : CategoryTheory.Comonad C) [U.Additive] (F : U.Coalgebra) (G : U.Coalgebra) (α : F ⟶ G) : (-α).f = -α.f

@[simp]

theorem CategoryTheory.Comonad.coalgebraPreadditive_homGroup_zsmul_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (U : CategoryTheory.Comonad C) [U.Additive] (F : U.Coalgebra) (G : U.Coalgebra) (r : ℤ) (α : F ⟶ G) : (r • α).f = r • α.f

@[simp]

theorem CategoryTheory.Comonad.coalgebraPreadditive_homGroup_sub_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (U : CategoryTheory.Comonad C) [U.Additive] (F : U.Coalgebra) (G : U.Coalgebra) (α : F ⟶ G) (β : F ⟶ G) : (α - β).f = α.f - β.f

@[simp]

theorem CategoryTheory.Comonad.coalgebraPreadditive_homGroup_nsmul_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (U : CategoryTheory.Comonad C) [U.Additive] (F : U.Coalgebra) (G : U.Coalgebra) (n : ℕ) (α : F ⟶ G) : (n • α).f = n • α.f

@[simp]

theorem CategoryTheory.Comonad.coalgebraPreadditive_homGroup_add_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (U : CategoryTheory.Comonad C) [U.Additive] (F : U.Coalgebra) (G : U.Coalgebra) (α : F ⟶ G) (β : F ⟶ G) : (α + β).f = α.f + β.f

@[simp]

theorem CategoryTheory.Comonad.coalgebraPreadditive_homGroup_zero_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (U : CategoryTheory.Comonad C) [U.Additive] (F : U.Coalgebra) (G : U.Coalgebra) : CategoryTheory.Comonad.Coalgebra.Hom.f 0 = 0

instance CategoryTheory.Comonad.coalgebraPreadditive (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (U : CategoryTheory.Comonad C) [U.Additive] : CategoryTheory.Preadditive U.Coalgebra

The category of coalgebras over an additive comonad on a preadditive category is preadditive.

Equations

• CategoryTheory.Comonad.coalgebraPreadditive C U = { homGroup := fun (F G : U.Coalgebra) => AddCommGroup.mk ⋯, add_comp := ⋯, comp_add := ⋯ }

instance CategoryTheory.Comonad.forget_additive (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (U : CategoryTheory.Comonad C) [U.Additive] : U.forget.Additive

Equations

• ⋯ = ⋯