Homology of linear categories #

In this file, it is shown that if C is a R-linear category, then ShortComplex C is a R-linear category. Various homological notions are also shown to be linear.

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instance CategoryTheory.ShortComplex.instSMulHom {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} :

SMul R (S₁ ⟶ S₂)

Equations

CategoryTheory.ShortComplex.instSMulHom = { smul := fun (a : R) (φ : S₁ ⟶ S₂) => { τ₁ := a • φ.τ₁, τ₂ := a • φ.τ₂, τ₃ := a • φ.τ₃, comm₁₂ := ⋯, comm₂₃ := ⋯ } }

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@[simp]

theorem CategoryTheory.ShortComplex.smul_τ₁ {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} (a : R) (φ : S₁ ⟶ S₂) :

(a • φ).τ₁ = a • φ.τ₁

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@[simp]

theorem CategoryTheory.ShortComplex.smul_τ₂ {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} (a : R) (φ : S₁ ⟶ S₂) :

(a • φ).τ₂ = a • φ.τ₂

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@[simp]

theorem CategoryTheory.ShortComplex.smul_τ₃ {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} (a : R) (φ : S₁ ⟶ S₂) :

(a • φ).τ₃ = a • φ.τ₃

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instance CategoryTheory.ShortComplex.instModuleHom {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} :

Module R (S₁ ⟶ S₂)

Equations

CategoryTheory.ShortComplex.instModuleHom = Module.mk ⋯ ⋯

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instance CategoryTheory.ShortComplex.instLinear {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] :

CategoryTheory.Linear R (CategoryTheory.ShortComplex C)

Equations

CategoryTheory.ShortComplex.instLinear = { homModule := inferInstance, smul_comp := ⋯, comp_smul := ⋯ }

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theorem CategoryTheory.ShortComplex.LeftHomologyMapData.smul_φK {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData} {h₂ : S₂.LeftHomologyData} (γ : CategoryTheory.ShortComplex.LeftHomologyMapData φ h₁ h₂) (a : R) :

(γ.smul a).φK = a • γ.φK

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theorem CategoryTheory.ShortComplex.LeftHomologyMapData.smul_φH {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData} {h₂ : S₂.LeftHomologyData} (γ : CategoryTheory.ShortComplex.LeftHomologyMapData φ h₁ h₂) (a : R) :

(γ.smul a).φH = a • γ.φH

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def CategoryTheory.ShortComplex.LeftHomologyMapData.smul {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData} {h₂ : S₂.LeftHomologyData} (γ : CategoryTheory.ShortComplex.LeftHomologyMapData φ h₁ h₂) (a : R) :

CategoryTheory.ShortComplex.LeftHomologyMapData (a • φ) h₁ h₂

Given a left homology map data for morphism φ, this is the induced left homology map data for a • φ.

Equations

γ.smul a = { φK := a • γ.φK, φH := a • γ.φH, commi := ⋯, commf' := ⋯, commπ := ⋯ }

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theorem CategoryTheory.ShortComplex.leftHomologyMap'_smul {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData) (a : R) :

CategoryTheory.ShortComplex.leftHomologyMap' (a • φ) h₁ h₂ = a • CategoryTheory.ShortComplex.leftHomologyMap' φ h₁ h₂

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@[simp]

theorem CategoryTheory.ShortComplex.cyclesMap'_smul {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData) (a : R) :

CategoryTheory.ShortComplex.cyclesMap' (a • φ) h₁ h₂ = a • CategoryTheory.ShortComplex.cyclesMap' φ h₁ h₂

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theorem CategoryTheory.ShortComplex.leftHomologyMap_smul {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (a : R) [S₁.HasLeftHomology] [S₂.HasLeftHomology] :

CategoryTheory.ShortComplex.leftHomologyMap (a • φ) = a • CategoryTheory.ShortComplex.leftHomologyMap φ

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theorem CategoryTheory.ShortComplex.cyclesMap_smul {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (a : R) [S₁.HasLeftHomology] [S₂.HasLeftHomology] :

CategoryTheory.ShortComplex.cyclesMap (a • φ) = a • CategoryTheory.ShortComplex.cyclesMap φ

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instance CategoryTheory.ShortComplex.leftHomologyFunctor_linear {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] :

CategoryTheory.Functor.Linear R (CategoryTheory.ShortComplex.leftHomologyFunctor C)

Equations

⋯ = ⋯

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instance CategoryTheory.ShortComplex.cyclesFunctor_linear {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] :

CategoryTheory.Functor.Linear R (CategoryTheory.ShortComplex.cyclesFunctor C)

Equations

⋯ = ⋯

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theorem CategoryTheory.ShortComplex.RightHomologyMapData.smul_φQ {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} (γ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂) (a : R) :

(γ.smul a).φQ = a • γ.φQ

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theorem CategoryTheory.ShortComplex.RightHomologyMapData.smul_φH {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} (γ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂) (a : R) :

(γ.smul a).φH = a • γ.φH

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def CategoryTheory.ShortComplex.RightHomologyMapData.smul {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} (γ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂) (a : R) :

CategoryTheory.ShortComplex.RightHomologyMapData (a • φ) h₁ h₂

Given a right homology map data for morphism φ, this is the induced right homology map data for a • φ.

Equations

γ.smul a = { φQ := a • γ.φQ, φH := a • γ.φH, commp := ⋯, commg' := ⋯, commι := ⋯ }

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theorem CategoryTheory.ShortComplex.rightHomologyMap'_smul {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData) (a : R) :

CategoryTheory.ShortComplex.rightHomologyMap' (a • φ) h₁ h₂ = a • CategoryTheory.ShortComplex.rightHomologyMap' φ h₁ h₂

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theorem CategoryTheory.ShortComplex.opcyclesMap'_smul {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData) (a : R) :

CategoryTheory.ShortComplex.opcyclesMap' (a • φ) h₁ h₂ = a • CategoryTheory.ShortComplex.opcyclesMap' φ h₁ h₂

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theorem CategoryTheory.ShortComplex.rightHomologyMap_smul {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (a : R) [S₁.HasRightHomology] [S₂.HasRightHomology] :

CategoryTheory.ShortComplex.rightHomologyMap (a • φ) = a • CategoryTheory.ShortComplex.rightHomologyMap φ

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theorem CategoryTheory.ShortComplex.opcyclesMap_smul {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (a : R) [S₁.HasRightHomology] [S₂.HasRightHomology] :

CategoryTheory.ShortComplex.opcyclesMap (a • φ) = a • CategoryTheory.ShortComplex.opcyclesMap φ

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instance CategoryTheory.ShortComplex.rightHomologyFunctor_linear {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] :

CategoryTheory.Functor.Linear R (CategoryTheory.ShortComplex.rightHomologyFunctor C)

Equations

⋯ = ⋯

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instance CategoryTheory.ShortComplex.opcyclesFunctor_linear {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] :

CategoryTheory.Functor.Linear R (CategoryTheory.ShortComplex.opcyclesFunctor C)

Equations

⋯ = ⋯

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theorem CategoryTheory.ShortComplex.HomologyMapData.smul_left {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData} (γ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂) (a : R) :

(γ.smul a).left = γ.left.smul a

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theorem CategoryTheory.ShortComplex.HomologyMapData.smul_right {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData} (γ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂) (a : R) :

(γ.smul a).right = γ.right.smul a

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def CategoryTheory.ShortComplex.HomologyMapData.smul {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData} (γ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂) (a : R) :

CategoryTheory.ShortComplex.HomologyMapData (a • φ) h₁ h₂

Given a homology map data for a morphism φ, this is the induced homology map data for a • φ.

Equations

γ.smul a = { left := γ.left.smul a, right := γ.right.smul a }

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@[simp]

theorem CategoryTheory.ShortComplex.homologyMap'_smul {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} (h₁ : S₁.HomologyData) (h₂ : S₂.HomologyData) (a : R) :

CategoryTheory.ShortComplex.homologyMap' (a • φ) h₁ h₂ = a • CategoryTheory.ShortComplex.homologyMap' φ h₁ h₂

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theorem CategoryTheory.ShortComplex.homologyMap_smul {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (a : R) [S₁.HasHomology] [S₂.HasHomology] :

CategoryTheory.ShortComplex.homologyMap (a • φ) = a • CategoryTheory.ShortComplex.homologyMap φ

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instance CategoryTheory.ShortComplex.homologyFunctor_linear {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] [CategoryTheory.CategoryWithHomology C] :

CategoryTheory.Functor.Linear R (CategoryTheory.ShortComplex.homologyFunctor C)

Equations

⋯ = ⋯

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theorem CategoryTheory.ShortComplex.Homotopy.smul_h₃ {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} {φ₁ : S₁ ⟶ S₂} {φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (a : R) :

(h.smul a).h₃ = a • h.h₃

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theorem CategoryTheory.ShortComplex.Homotopy.smul_h₀ {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} {φ₁ : S₁ ⟶ S₂} {φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (a : R) :

(h.smul a).h₀ = a • h.h₀

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@[simp]

theorem CategoryTheory.ShortComplex.Homotopy.smul_h₂ {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} {φ₁ : S₁ ⟶ S₂} {φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (a : R) :

(h.smul a).h₂ = a • h.h₂

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@[simp]

theorem CategoryTheory.ShortComplex.Homotopy.smul_h₁ {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} {φ₁ : S₁ ⟶ S₂} {φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (a : R) :

(h.smul a).h₁ = a • h.h₁

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def CategoryTheory.ShortComplex.Homotopy.smul {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ : CategoryTheory.ShortComplex C} {S₂ : CategoryTheory.ShortComplex C} {φ₁ : S₁ ⟶ S₂} {φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (a : R) :

CategoryTheory.ShortComplex.Homotopy (a • φ₁) (a • φ₂)

Homotopy between morphisms of short complexes is compatible with the scalar multiplication.

Equations

h.smul a = { h₀ := a • h.h₀, h₀_f := ⋯, h₁ := a • h.h₁, h₂ := a • h.h₂, h₃ := a • h.h₃, g_h₃ := ⋯, comm₁ := ⋯, comm₂ := ⋯, comm₃ := ⋯ }

Instances For

Documentation

Mathlib.Algebra.Homology.ShortComplex.Linear

Homology of linear categories #