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

(a • φ).τ₁ = a • φ.τ₁

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

Module R (S₁ ⟶ S₂)

Equations

CategoryTheory.ShortComplex.instModuleHom = { toSMul := CategoryTheory.ShortComplex.instSMulHom, one_smul := ⋯, mul_smul := ⋯, smul_zero := ⋯, smul_add := ⋯, add_smul := ⋯, zero_smul := ⋯ }

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

Linear R (ShortComplex C)

Equations

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

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

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

(γ.smul a).φH = a • γ.φH

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

(γ.smul a).φK = a • γ.φK

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

leftHomologyMap' (a • φ) h₁ h₂ = a • leftHomologyMap' φ h₁ h₂

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

cyclesMap' (a • φ) h₁ h₂ = a • cyclesMap' φ h₁ h₂

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

leftHomologyMap (a • φ) = a • leftHomologyMap φ

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

cyclesMap (a • φ) = a • cyclesMap φ

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

Functor.Linear R (leftHomologyFunctor C)

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

Functor.Linear R (cyclesFunctor C)

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

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

(γ.smul a).φH = a • γ.φH

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

(γ.smul a).φQ = a • γ.φQ

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@[simp]

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

rightHomologyMap' (a • φ) h₁ h₂ = a • rightHomologyMap' φ h₁ h₂

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

opcyclesMap' (a • φ) h₁ h₂ = a • opcyclesMap' φ h₁ h₂

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

rightHomologyMap (a • φ) = a • rightHomologyMap φ

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

opcyclesMap (a • φ) = a • opcyclesMap φ

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

Functor.Linear R (rightHomologyFunctor C)

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

Functor.Linear R (opcyclesFunctor C)

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

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

(γ.smul a).right = γ.right.smul a

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

(γ.smul a).left = γ.left.smul a

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

homologyMap' (a • φ) h₁ h₂ = a • homologyMap' φ h₁ h₂

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

homologyMap (a • φ) = a • homologyMap φ

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

Functor.Linear R (homologyFunctor C)

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

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 := ⋯ }

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

(h.smul a).h₂ = a • h.h₂

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@[simp]

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

(h.smul a).h₀ = a • h.h₀

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

(h.smul a).h₃ = a • h.h₃

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

(h.smul a).h₁ = a • h.h₁

Documentation

Mathlib.Algebra.Homology.ShortComplex.Linear

Homology of linear categories #