The shift on cochain complexes and on the homotopy category

In this file, we show that for any preadditive category C, the categories CochainComplex C ℤ and HomotopyCategory C (ComplexShape.up ℤ) are equipped with a shift by ℤ.

We also show that if F : C ⥤ D is an additive functor, then the functors F.mapHomologicalComplex (ComplexShape.up ℤ) and F.mapHomotopyCategory (ComplexShape.up ℤ) commute with the shift by ℤ.

def CochainComplex.shiftFunctor (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (n : ℤ) : CategoryTheory.Functor (CochainComplex C ℤ) (CochainComplex C ℤ)

The shift functor by n : ℤ on CochainComplex C ℤ which sends a cochain complex K to the complex which is K.X (i + n) in degree i, and which multiplies the differentials by (-1)^n.

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theorem CochainComplex.shiftFunctor_map_f (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (n : ℤ) {X✝ Y✝ : CochainComplex C ℤ} (φ : X✝ ⟶ Y✝) (x✝ : ℤ) : ((CochainComplex.shiftFunctor C n).map φ).f x✝ = φ.f (x✝ + n)

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theorem CochainComplex.shiftFunctor_obj_X (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (n : ℤ) (K : CochainComplex C ℤ) (i : ℤ) : ((CochainComplex.shiftFunctor C n).obj K).X i = K.X (i + n)

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theorem CochainComplex.shiftFunctor_obj_d (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (n : ℤ) (K : CochainComplex C ℤ) (x✝ x✝¹ : ℤ) : ((CochainComplex.shiftFunctor C n).obj K).d x✝ x✝¹ = n.negOnePow • K.d (x✝ + n) (x✝¹ + n)

instance CochainComplex.instAdditiveIntShiftFunctor (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (n : ℤ) : (CochainComplex.shiftFunctor C n).Additive

def CochainComplex.shiftFunctorObjXIso {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (K : CochainComplex C ℤ) (n i m : ℤ) (hm : m = i + n) : ((CochainComplex.shiftFunctor C n).obj K).X i ≅ K.X m

The canonical isomorphism ((shiftFunctor C n).obj K).X i ≅ K.X m when m = i + n.

Equations

• K.shiftFunctorObjXIso n i m hm = HomologicalComplex.XIsoOfEq K ⋯

def CochainComplex.shiftFunctorZero' (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (n : ℤ) (h : n = 0) : CochainComplex.shiftFunctor C n ≅ CategoryTheory.Functor.id (CochainComplex C ℤ)

The shift functor by n on CochainComplex C ℤ identifies to the identity functor when n = 0.

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theorem CochainComplex.shiftFunctorZero'_hom_app_f (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (n : ℤ) (h : n = 0) (X : CochainComplex C ℤ) (i : ℤ) : ((CochainComplex.shiftFunctorZero' C n h).hom.app X).f i = (HomologicalComplex.XIsoOfEq X ⋯).hom

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theorem CochainComplex.shiftFunctorZero'_inv_app_f (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (n : ℤ) (h : n = 0) (X : CochainComplex C ℤ) (i : ℤ) : ((CochainComplex.shiftFunctorZero' C n h).inv.app X).f i = (HomologicalComplex.XIsoOfEq X ⋯).inv

def CochainComplex.shiftFunctorAdd' (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (n₁ n₂ n₁₂ : ℤ) (h : n₁ + n₂ = n₁₂) : CochainComplex.shiftFunctor C n₁₂ ≅ (CochainComplex.shiftFunctor C n₁).comp (CochainComplex.shiftFunctor C n₂)

The compatibility of the shift functors on CochainComplex C ℤ with respect to the addition of integers.

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theorem CochainComplex.shiftFunctorAdd'_hom_app_f (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (n₁ n₂ n₁₂ : ℤ) (h : n₁ + n₂ = n₁₂) (X : CochainComplex C ℤ) (i : ℤ) : ((CochainComplex.shiftFunctorAdd' C n₁ n₂ n₁₂ h).hom.app X).f i = (HomologicalComplex.XIsoOfEq X ⋯).hom

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theorem CochainComplex.shiftFunctorAdd'_inv_app_f (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (n₁ n₂ n₁₂ : ℤ) (h : n₁ + n₂ = n₁₂) (X : CochainComplex C ℤ) (i : ℤ) : ((CochainComplex.shiftFunctorAdd' C n₁ n₂ n₁₂ h).inv.app X).f i = (HomologicalComplex.XIsoOfEq X ⋯).inv

instance CochainComplex.instHasShiftInt (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] : CategoryTheory.HasShift (CochainComplex C ℤ) ℤ

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instance CochainComplex.instAdditiveHomologicalComplexIntUpShiftFunctor (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (n : ℤ) : (CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).Additive

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theorem CochainComplex.shiftFunctor_obj_X' {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (K : CochainComplex C ℤ) (n p : ℤ) : ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K).X p = K.X (p + n)

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theorem CochainComplex.shiftFunctor_map_f' {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} (φ : K ⟶ L) (n p : ℤ) : ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).map φ).f p = φ.f (p + n)

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theorem CochainComplex.shiftFunctor_obj_d' {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (K : CochainComplex C ℤ) (n i j : ℤ) : ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K).d i j = n.negOnePow • K.d (i + { as := n }.as) (j + { as := n }.as)

theorem CochainComplex.shiftFunctorAdd_inv_app_f {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (K : CochainComplex C ℤ) (a b n : ℤ) : ((CategoryTheory.shiftFunctorAdd (CochainComplex C ℤ) a b).inv.app K).f n = (HomologicalComplex.XIsoOfEq K ⋯).hom

theorem CochainComplex.shiftFunctorAdd_hom_app_f {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (K : CochainComplex C ℤ) (a b n : ℤ) : ((CategoryTheory.shiftFunctorAdd (CochainComplex C ℤ) a b).hom.app K).f n = (HomologicalComplex.XIsoOfEq K ⋯).hom

theorem CochainComplex.shiftFunctorAdd'_inv_app_f' {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (K : CochainComplex C ℤ) (a b ab : ℤ) (h : a + b = ab) (n : ℤ) : ((CategoryTheory.shiftFunctorAdd' (CochainComplex C ℤ) a b ab h).inv.app K).f n = (HomologicalComplex.XIsoOfEq K ⋯).hom

theorem CochainComplex.shiftFunctorAdd'_hom_app_f' {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (K : CochainComplex C ℤ) (a b ab : ℤ) (h : a + b = ab) (n : ℤ) : ((CategoryTheory.shiftFunctorAdd' (CochainComplex C ℤ) a b ab h).hom.app K).f n = (HomologicalComplex.XIsoOfEq K ⋯).hom

theorem CochainComplex.shiftFunctorZero_inv_app_f {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (K : CochainComplex C ℤ) (n : ℤ) : ((CategoryTheory.shiftFunctorZero (CochainComplex C ℤ) ℤ).inv.app K).f n = (HomologicalComplex.XIsoOfEq K ⋯).hom

theorem CochainComplex.shiftFunctorZero_hom_app_f {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (K : CochainComplex C ℤ) (n : ℤ) : ((CategoryTheory.shiftFunctorZero (CochainComplex C ℤ) ℤ).hom.app K).f n = (HomologicalComplex.XIsoOfEq K ⋯).hom

theorem CochainComplex.XIsoOfEq_shift {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (K : CochainComplex C ℤ) (n : ℤ) {p q : ℤ} (hpq : p = q) : HomologicalComplex.XIsoOfEq ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K) hpq = HomologicalComplex.XIsoOfEq K ⋯

theorem CochainComplex.shiftFunctorAdd'_eq (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (a b c : ℤ) (h : a + b = c) : CategoryTheory.shiftFunctorAdd' (CochainComplex C ℤ) a b c h = CochainComplex.shiftFunctorAdd' C a b c h

theorem CochainComplex.shiftFunctorAdd_eq (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (a b : ℤ) : CategoryTheory.shiftFunctorAdd (CochainComplex C ℤ) a b = CochainComplex.shiftFunctorAdd' C a b (a + b) ⋯

theorem CochainComplex.shiftFunctorZero_eq (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] : CategoryTheory.shiftFunctorZero (CochainComplex C ℤ) ℤ = CochainComplex.shiftFunctorZero' C 0 ⋯

theorem CochainComplex.shiftFunctorComm_hom_app_f {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (K : CochainComplex C ℤ) (a b p : ℤ) : ((CategoryTheory.shiftFunctorComm (CochainComplex C ℤ) a b).hom.app K).f p = (HomologicalComplex.XIsoOfEq K ⋯).hom

def CochainComplex.shiftEval (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (n i i' : ℤ) (hi : n + i = i') : (CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).comp (HomologicalComplex.eval C (ComplexShape.up ℤ) i) ≅ HomologicalComplex.eval C (ComplexShape.up ℤ) i'

Shifting cochain complexes by n and evaluating in a degree i identifies to the evaluation in degree i' when n + i = i'.

Equations

• CochainComplex.shiftEval C n i i' hi = CategoryTheory.NatIso.ofComponents (fun (K : CochainComplex C ℤ) => HomologicalComplex.XIsoOfEq K ⋯) ⋯

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theorem CochainComplex.shiftEval_inv_app (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (n i i' : ℤ) (hi : n + i = i') (X : CochainComplex C ℤ) : (CochainComplex.shiftEval C n i i' hi).inv.app X = (HomologicalComplex.XIsoOfEq X ⋯).inv

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theorem CochainComplex.shiftEval_hom_app (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (n i i' : ℤ) (hi : n + i = i') (X : CochainComplex C ℤ) : (CochainComplex.shiftEval C n i i' hi).hom.app X = (HomologicalComplex.XIsoOfEq X ⋯).hom

def CategoryTheory.Functor.mapCochainComplexShiftIso {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] [CategoryTheory.Preadditive D] (F : CategoryTheory.Functor C D) [F.Additive] (n : ℤ) : (CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).comp (F.mapHomologicalComplex (ComplexShape.up ℤ)) ≅ (F.mapHomologicalComplex (ComplexShape.up ℤ)).comp (CategoryTheory.shiftFunctor (HomologicalComplex D (ComplexShape.up ℤ)) n)

The commutation with the shift isomorphism for the functor on cochain complexes induced by an additive functor between preadditive categories.

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theorem CategoryTheory.Functor.mapCochainComplexShiftIso_inv_app_f {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] [CategoryTheory.Preadditive D] (F : CategoryTheory.Functor C D) [F.Additive] (n : ℤ) (X : HomologicalComplex C (ComplexShape.up ℤ)) (i : ℤ) : ((F.mapCochainComplexShiftIso n).inv.app X).f i = CategoryTheory.CategoryStruct.id (F.obj (X.X (i + n)))

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theorem CategoryTheory.Functor.mapCochainComplexShiftIso_hom_app_f {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] [CategoryTheory.Preadditive D] (F : CategoryTheory.Functor C D) [F.Additive] (n : ℤ) (X : HomologicalComplex C (ComplexShape.up ℤ)) (i : ℤ) : ((F.mapCochainComplexShiftIso n).hom.app X).f i = CategoryTheory.CategoryStruct.id (F.obj (X.X (i + n)))

instance CategoryTheory.Functor.commShiftMapCochainComplex {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] [CategoryTheory.Preadditive D] (F : CategoryTheory.Functor C D) [F.Additive] : (F.mapHomologicalComplex (ComplexShape.up ℤ)).CommShift ℤ

Equations

• F.commShiftMapCochainComplex = { iso := F.mapCochainComplexShiftIso, zero := ⋯, add := ⋯ }

theorem CategoryTheory.Functor.mapHomologicalComplex_commShiftIso_eq {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] [CategoryTheory.Preadditive D] (F : CategoryTheory.Functor C D) [F.Additive] (n : ℤ) : (F.mapHomologicalComplex (ComplexShape.up ℤ)).commShiftIso n = F.mapCochainComplexShiftIso n

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theorem CategoryTheory.Functor.mapHomologicalComplex_commShiftIso_hom_app_f {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] [CategoryTheory.Preadditive D] (F : CategoryTheory.Functor C D) [F.Additive] (K : CochainComplex C ℤ) (n i : ℤ) : (((F.mapHomologicalComplex (ComplexShape.up ℤ)).commShiftIso n).hom.app K).f i = CategoryTheory.CategoryStruct.id ((((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).comp (F.mapHomologicalComplex (ComplexShape.up ℤ))).obj K).X i)

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theorem CategoryTheory.Functor.mapHomologicalComplex_commShiftIso_inv_app_f {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] [CategoryTheory.Preadditive D] (F : CategoryTheory.Functor C D) [F.Additive] (K : CochainComplex C ℤ) (n i : ℤ) : (((F.mapHomologicalComplex (ComplexShape.up ℤ)).commShiftIso n).inv.app K).f i = CategoryTheory.CategoryStruct.id ((((F.mapHomologicalComplex (ComplexShape.up ℤ)).comp (CategoryTheory.shiftFunctor (HomologicalComplex D (ComplexShape.up ℤ)) n)).obj K).X i)

def Homotopy.shift {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} {φ₁ φ₂ : K ⟶ L} (h : Homotopy φ₁ φ₂) (n : ℤ) : Homotopy ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).map φ₁) ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).map φ₂)

If h : Homotopy φ₁ φ₂ and n : ℤ, this is the induced homotopy between φ₁⟦n⟧' and φ₂⟦n⟧'.

Equations

• h.shift n = { hom := fun (x x_1 : ℤ) => n.negOnePow • h.hom (x + { as := n }.as) (x_1 + { as := n }.as), zero := ⋯, comm := ⋯ }

instance HomotopyCategory.instIsCompatibleWithShiftHomologicalComplexIntUpHomotopic (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] : (homotopic C (ComplexShape.up ℤ)).IsCompatibleWithShift ℤ

noncomputable instance HomotopyCategory.hasShift (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] : CategoryTheory.HasShift (HomotopyCategory C (ComplexShape.up ℤ)) ℤ

Equations

• HomotopyCategory.hasShift C = id inferInstance

noncomputable instance HomotopyCategory.commShiftQuotient (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] : (HomotopyCategory.quotient C (ComplexShape.up ℤ)).CommShift ℤ

Equations

• HomotopyCategory.commShiftQuotient C = CategoryTheory.Quotient.functor_commShift (homotopic C (ComplexShape.up ℤ)) ℤ

instance HomotopyCategory.instAdditiveIntUpShiftFunctor (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (n : ℤ) : (CategoryTheory.shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).Additive

noncomputable instance HomotopyCategory.instCommShiftIntUpMapHomotopyCategory {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] [CategoryTheory.Preadditive D] (F : CategoryTheory.Functor C D) [F.Additive] : (F.mapHomotopyCategory (ComplexShape.up ℤ)).CommShift ℤ

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• One or more equations did not get rendered due to their size.

instance HomotopyCategory.instCommShiftHomologicalComplexIntUpHomFunctorMapHomotopyCategoryFactors {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] [CategoryTheory.Preadditive D] (F : CategoryTheory.Functor C D) [F.Additive] : CategoryTheory.NatTrans.CommShift (F.mapHomotopyCategoryFactors (ComplexShape.up ℤ)).hom ℤ