Preadditive structure on algebras over a monad

If C is a preadditive category and F is an additive endofunctor on C then Algebra F is also preadditive. Dually, the category Coalgebra F is also preadditive.

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theorem CategoryTheory.Endofunctor.algebraPreadditive_homGroup_add_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (F : CategoryTheory.Functor C C) [F.Additive] (A₁ : CategoryTheory.Endofunctor.Algebra F) (A₂ : CategoryTheory.Endofunctor.Algebra F) (α : A₁ ⟶ A₂) (β : A₁ ⟶ A₂) : (α + β).f = α.f + β.f

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theorem CategoryTheory.Endofunctor.algebraPreadditive_homGroup_sub_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (F : CategoryTheory.Functor C C) [F.Additive] (A₁ : CategoryTheory.Endofunctor.Algebra F) (A₂ : CategoryTheory.Endofunctor.Algebra F) (α : A₁ ⟶ A₂) (β : A₁ ⟶ A₂) : (α - β).f = α.f - β.f

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theorem CategoryTheory.Endofunctor.algebraPreadditive_homGroup_zero_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (F : CategoryTheory.Functor C C) [F.Additive] (A₁ : CategoryTheory.Endofunctor.Algebra F) (A₂ : CategoryTheory.Endofunctor.Algebra F) : CategoryTheory.Endofunctor.Algebra.Hom.f 0 = 0

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theorem CategoryTheory.Endofunctor.algebraPreadditive_homGroup_neg_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (F : CategoryTheory.Functor C C) [F.Additive] (A₁ : CategoryTheory.Endofunctor.Algebra F) (A₂ : CategoryTheory.Endofunctor.Algebra F) (α : A₁ ⟶ A₂) : (-α).f = -α.f

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theorem CategoryTheory.Endofunctor.algebraPreadditive_homGroup_zsmul_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (F : CategoryTheory.Functor C C) [F.Additive] (A₁ : CategoryTheory.Endofunctor.Algebra F) (A₂ : CategoryTheory.Endofunctor.Algebra F) (r : ℤ) (α : A₁ ⟶ A₂) : (r • α).f = r • α.f

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theorem CategoryTheory.Endofunctor.algebraPreadditive_homGroup_nsmul_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (F : CategoryTheory.Functor C C) [F.Additive] (A₁ : CategoryTheory.Endofunctor.Algebra F) (A₂ : CategoryTheory.Endofunctor.Algebra F) (n : ℕ) (α : A₁ ⟶ A₂) : (n • α).f = n • α.f

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

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

Equations

• CategoryTheory.Endofunctor.algebraPreadditive C F = { homGroup := fun (A₁ A₂ : CategoryTheory.Endofunctor.Algebra F) => AddCommGroup.mk ⋯, add_comp := ⋯, comp_add := ⋯ }

instance CategoryTheory.Algebra.forget_additive (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (F : CategoryTheory.Functor C C) [F.Additive] : (CategoryTheory.Endofunctor.Algebra.forget F).Additive

Equations

• ⋯ = ⋯

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theorem CategoryTheory.Endofunctor.coalgebraPreadditive_homGroup_sub_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (F : CategoryTheory.Functor C C) [F.Additive] (A₁ : CategoryTheory.Endofunctor.Coalgebra F) (A₂ : CategoryTheory.Endofunctor.Coalgebra F) (α : A₁ ⟶ A₂) (β : A₁ ⟶ A₂) : (α - β).f = α.f - β.f

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theorem CategoryTheory.Endofunctor.coalgebraPreadditive_homGroup_add_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (F : CategoryTheory.Functor C C) [F.Additive] (A₁ : CategoryTheory.Endofunctor.Coalgebra F) (A₂ : CategoryTheory.Endofunctor.Coalgebra F) (α : A₁ ⟶ A₂) (β : A₁ ⟶ A₂) : (α + β).f = α.f + β.f

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theorem CategoryTheory.Endofunctor.coalgebraPreadditive_homGroup_zero_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (F : CategoryTheory.Functor C C) [F.Additive] (A₁ : CategoryTheory.Endofunctor.Coalgebra F) (A₂ : CategoryTheory.Endofunctor.Coalgebra F) : CategoryTheory.Endofunctor.Coalgebra.Hom.f 0 = 0

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theorem CategoryTheory.Endofunctor.coalgebraPreadditive_homGroup_neg_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (F : CategoryTheory.Functor C C) [F.Additive] (A₁ : CategoryTheory.Endofunctor.Coalgebra F) (A₂ : CategoryTheory.Endofunctor.Coalgebra F) (α : A₁ ⟶ A₂) : (-α).f = -α.f

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theorem CategoryTheory.Endofunctor.coalgebraPreadditive_homGroup_zsmul_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (F : CategoryTheory.Functor C C) [F.Additive] (A₁ : CategoryTheory.Endofunctor.Coalgebra F) (A₂ : CategoryTheory.Endofunctor.Coalgebra F) (r : ℤ) (α : A₁ ⟶ A₂) : (r • α).f = r • α.f

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theorem CategoryTheory.Endofunctor.coalgebraPreadditive_homGroup_nsmul_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (F : CategoryTheory.Functor C C) [F.Additive] (A₁ : CategoryTheory.Endofunctor.Coalgebra F) (A₂ : CategoryTheory.Endofunctor.Coalgebra F) (n : ℕ) (α : A₁ ⟶ A₂) : (n • α).f = n • α.f

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

Equations

• CategoryTheory.Endofunctor.coalgebraPreadditive C F = { homGroup := fun (A₁ A₂ : CategoryTheory.Endofunctor.Coalgebra F) => AddCommGroup.mk ⋯, add_comp := ⋯, comp_add := ⋯ }

instance CategoryTheory.Coalgebra.forget_additive (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] (F : CategoryTheory.Functor C C) [F.Additive] : (CategoryTheory.Endofunctor.Coalgebra.forget F).Additive

Equations

• ⋯ = ⋯