Opposite categories of complexes

Given a preadditive category V, the opposite of its category of chain complexes is equivalent to the category of cochain complexes of objects in Vᵒᵖ. We define this equivalence, and another analogous equivalence (for a general category of homological complexes with a general complex shape).

We then show that when V is abelian, if C is a homological complex, then the homology of op(C) is isomorphic to op of the homology of C (and the analogous result for unop).

Implementation notes

It is convenient to define both op and opSymm; this is because given a complex shape c, c.symm.symm is not defeq to c.

Tags

opposite, chain complex, cochain complex, homology, cohomology, homological complex

theorem imageToKernel_op {V : Type u_1} [CategoryTheory.Category.{u_2, u_1} V] [CategoryTheory.Abelian V] {X : V} {Y : V} {Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : CategoryTheory.CategoryStruct.comp f g = 0) : imageToKernel g.op f.op ⋯ = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.imageSubobjectIso g.op ≪≫ (CategoryTheory.imageOpOp g).symm).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.cokernel.desc f (CategoryTheory.Limits.factorThruImage g) ⋯).op (CategoryTheory.Limits.kernelSubobjectIso f.op ≪≫ CategoryTheory.kernelOpOp f).inv)

theorem imageToKernel_unop {V : Type u_1} [CategoryTheory.Category.{u_2, u_1} V] [CategoryTheory.Abelian V] {X : Vᵒᵖ} {Y : Vᵒᵖ} {Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : CategoryTheory.CategoryStruct.comp f g = 0) : imageToKernel g.unop f.unop ⋯ = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.imageSubobjectIso g.unop ≪≫ (CategoryTheory.imageUnopUnop g).symm).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.cokernel.desc f (CategoryTheory.Limits.factorThruImage g) ⋯).unop (CategoryTheory.Limits.kernelSubobjectIso f.unop ≪≫ CategoryTheory.kernelUnopUnop f).inv)

def homology'Op {V : Type u_1} [CategoryTheory.Category.{u_2, u_1} V] [CategoryTheory.Abelian V] {X : V} {Y : V} {Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : CategoryTheory.CategoryStruct.comp f g = 0) : homology' g.op f.op ⋯ ≅ { unop := homology' f g w }

Given f, g with f ≫ g = 0, the homology of g.op, f.op is the opposite of the homology of f, g.

Equations

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

def homology'Unop {V : Type u_1} [CategoryTheory.Category.{u_2, u_1} V] [CategoryTheory.Abelian V] {X : Vᵒᵖ} {Y : Vᵒᵖ} {Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : CategoryTheory.CategoryStruct.comp f g = 0) : homology' g.unop f.unop ⋯ ≅ (homology' f g w).unop

Given morphisms f, g in Vᵒᵖ with f ≫ g = 0, the homology of g.unop, f.unop is the opposite of the homology of f, g.

Equations

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

@[simp]

theorem HomologicalComplex.op_d {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Preadditive V] (X : HomologicalComplex V c) (i : ι) (j : ι) : X.op.d i j = (X.d j i).op

@[simp]

theorem HomologicalComplex.op_X {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Preadditive V] (X : HomologicalComplex V c) (i : ι) : X.op.X i = { unop := X.X i }

def HomologicalComplex.op {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Preadditive V] (X : HomologicalComplex V c) : HomologicalComplex Vᵒᵖ c.symm

Sends a complex X with objects in V to the corresponding complex with objects in Vᵒᵖ.

Equations

• X.op = { X := fun (i : ι) => { unop := X.X i }, d := fun (i j : ι) => (X.d j i).op, shape := ⋯, d_comp_d' := ⋯ }

@[simp]

theorem HomologicalComplex.opSymm_X {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Preadditive V] (X : HomologicalComplex V c.symm) (i : ι) : X.opSymm.X i = { unop := X.X i }

@[simp]

theorem HomologicalComplex.opSymm_d {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Preadditive V] (X : HomologicalComplex V c.symm) (i : ι) (j : ι) : X.opSymm.d i j = (X.d j i).op

def HomologicalComplex.opSymm {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Preadditive V] (X : HomologicalComplex V c.symm) : HomologicalComplex Vᵒᵖ c

Sends a complex X with objects in V to the corresponding complex with objects in Vᵒᵖ.

Equations

• X.opSymm = { X := fun (i : ι) => { unop := X.X i }, d := fun (i j : ι) => (X.d j i).op, shape := ⋯, d_comp_d' := ⋯ }

@[simp]

theorem HomologicalComplex.unop_d {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Preadditive V] (X : HomologicalComplex Vᵒᵖ c) (i : ι) (j : ι) : X.unop.d i j = (X.d j i).unop

@[simp]

theorem HomologicalComplex.unop_X {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Preadditive V] (X : HomologicalComplex Vᵒᵖ c) (i : ι) : X.unop.X i = (X.X i).unop

def HomologicalComplex.unop {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Preadditive V] (X : HomologicalComplex Vᵒᵖ c) : HomologicalComplex V c.symm

Sends a complex X with objects in Vᵒᵖ to the corresponding complex with objects in V.

Equations

• X.unop = { X := fun (i : ι) => (X.X i).unop, d := fun (i j : ι) => (X.d j i).unop, shape := ⋯, d_comp_d' := ⋯ }

@[simp]

theorem HomologicalComplex.unopSymm_d {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Preadditive V] (X : HomologicalComplex Vᵒᵖ c.symm) (i : ι) (j : ι) : X.unopSymm.d i j = (X.d j i).unop

@[simp]

theorem HomologicalComplex.unopSymm_X {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Preadditive V] (X : HomologicalComplex Vᵒᵖ c.symm) (i : ι) : X.unopSymm.X i = (X.X i).unop

def HomologicalComplex.unopSymm {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Preadditive V] (X : HomologicalComplex Vᵒᵖ c.symm) : HomologicalComplex V c

Sends a complex X with objects in Vᵒᵖ to the corresponding complex with objects in V.

Equations

• X.unopSymm = { X := fun (i : ι) => (X.X i).unop, d := fun (i j : ι) => (X.d j i).unop, shape := ⋯, d_comp_d' := ⋯ }

@[simp]

theorem HomologicalComplex.opFunctor_map_f {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : ∀ {X Y : (HomologicalComplex V c)ᵒᵖ} (f : X ⟶ Y) (i : ι), ((HomologicalComplex.opFunctor V c).map f).f i = (f.unop.f i).op

@[simp]

theorem HomologicalComplex.opFunctor_obj {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] (X : (HomologicalComplex V c)ᵒᵖ) : (HomologicalComplex.opFunctor V c).obj X = X.unop.op

def HomologicalComplex.opFunctor {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : CategoryTheory.Functor (HomologicalComplex V c)ᵒᵖ (HomologicalComplex Vᵒᵖ c.symm)

Auxiliary definition for opEquivalence.

Equations

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

@[simp]

theorem HomologicalComplex.opInverse_obj {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] (X : HomologicalComplex Vᵒᵖ c.symm) : (HomologicalComplex.opInverse V c).obj X = { unop := X.unopSymm }

@[simp]

theorem HomologicalComplex.opInverse_map {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : ∀ {X Y : HomologicalComplex Vᵒᵖ c.symm} (f : X ⟶ Y), (HomologicalComplex.opInverse V c).map f = Quiver.Hom.op { f := fun (i : ι) => (f.f i).unop, comm' := ⋯ }

def HomologicalComplex.opInverse {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : CategoryTheory.Functor (HomologicalComplex Vᵒᵖ c.symm) (HomologicalComplex V c)ᵒᵖ

Auxiliary definition for opEquivalence.

Equations

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

def HomologicalComplex.opUnitIso {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : CategoryTheory.Functor.id (HomologicalComplex V c)ᵒᵖ ≅ (HomologicalComplex.opFunctor V c).comp (HomologicalComplex.opInverse V c)

Auxiliary definition for opEquivalence.

Equations

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

def HomologicalComplex.opCounitIso {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : (HomologicalComplex.opInverse V c).comp (HomologicalComplex.opFunctor V c) ≅ CategoryTheory.Functor.id (HomologicalComplex Vᵒᵖ c.symm)

Auxiliary definition for opEquivalence.

Equations

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

@[simp]

theorem HomologicalComplex.opEquivalence_unitIso {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : (HomologicalComplex.opEquivalence V c).unitIso = HomologicalComplex.opUnitIso V c

@[simp]

theorem HomologicalComplex.opEquivalence_inverse {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : (HomologicalComplex.opEquivalence V c).inverse = HomologicalComplex.opInverse V c

@[simp]

theorem HomologicalComplex.opEquivalence_functor {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : (HomologicalComplex.opEquivalence V c).functor = HomologicalComplex.opFunctor V c

@[simp]

theorem HomologicalComplex.opEquivalence_counitIso {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : (HomologicalComplex.opEquivalence V c).counitIso = HomologicalComplex.opCounitIso V c

def HomologicalComplex.opEquivalence {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : (HomologicalComplex V c)ᵒᵖ ≌ HomologicalComplex Vᵒᵖ c.symm

Given a category of complexes with objects in V, there is a natural equivalence between its opposite category and a category of complexes with objects in Vᵒᵖ.

Equations

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

@[simp]

theorem HomologicalComplex.unopFunctor_map_f {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : ∀ {X Y : (HomologicalComplex Vᵒᵖ c)ᵒᵖ} (f : X ⟶ Y) (i : ι), ((HomologicalComplex.unopFunctor V c).map f).f i = (f.unop.f i).unop

@[simp]

theorem HomologicalComplex.unopFunctor_obj {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] (X : (HomologicalComplex Vᵒᵖ c)ᵒᵖ) : (HomologicalComplex.unopFunctor V c).obj X = X.unop.unop

def HomologicalComplex.unopFunctor {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : CategoryTheory.Functor (HomologicalComplex Vᵒᵖ c)ᵒᵖ (HomologicalComplex V c.symm)

Auxiliary definition for unopEquivalence.

Equations

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

@[simp]

theorem HomologicalComplex.unopInverse_map {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : ∀ {X Y : HomologicalComplex V c.symm} (f : X ⟶ Y), (HomologicalComplex.unopInverse V c).map f = Quiver.Hom.op { f := fun (i : ι) => (f.f i).op, comm' := ⋯ }

@[simp]

theorem HomologicalComplex.unopInverse_obj {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] (X : HomologicalComplex V c.symm) : (HomologicalComplex.unopInverse V c).obj X = { unop := X.opSymm }

def HomologicalComplex.unopInverse {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : CategoryTheory.Functor (HomologicalComplex V c.symm) (HomologicalComplex Vᵒᵖ c)ᵒᵖ

Auxiliary definition for unopEquivalence.

Equations

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

def HomologicalComplex.unopUnitIso {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : CategoryTheory.Functor.id (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≅ (HomologicalComplex.unopFunctor V c).comp (HomologicalComplex.unopInverse V c)

Auxiliary definition for unopEquivalence.

Equations

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

def HomologicalComplex.unopCounitIso {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : (HomologicalComplex.unopInverse V c).comp (HomologicalComplex.unopFunctor V c) ≅ CategoryTheory.Functor.id (HomologicalComplex V c.symm)

Auxiliary definition for unopEquivalence.

Equations

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

@[simp]

theorem HomologicalComplex.unopEquivalence_unitIso {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : (HomologicalComplex.unopEquivalence V c).unitIso = HomologicalComplex.unopUnitIso V c

@[simp]

theorem HomologicalComplex.unopEquivalence_counitIso {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : (HomologicalComplex.unopEquivalence V c).counitIso = HomologicalComplex.unopCounitIso V c

@[simp]

theorem HomologicalComplex.unopEquivalence_functor {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : (HomologicalComplex.unopEquivalence V c).functor = HomologicalComplex.unopFunctor V c

@[simp]

theorem HomologicalComplex.unopEquivalence_inverse {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : (HomologicalComplex.unopEquivalence V c).inverse = HomologicalComplex.unopInverse V c

def HomologicalComplex.unopEquivalence {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Preadditive V] : (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≌ HomologicalComplex V c.symm

Given a category of complexes with objects in Vᵒᵖ, there is a natural equivalence between its opposite category and a category of complexes with objects in V.

Equations

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

instance HomologicalComplex.opFunctor_additive {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Preadditive V] : (HomologicalComplex.opFunctor V c).Additive

Equations

• ⋯ = ⋯

instance HomologicalComplex.unopFunctor_additive {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Preadditive V] : (HomologicalComplex.unopFunctor V c).Additive

Equations

• ⋯ = ⋯

instance HomologicalComplex.instHasHomologyOppositeOp {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Preadditive V] (K : HomologicalComplex V c) (i : ι) [K.HasHomology i] : K.op.HasHomology i

Equations

• ⋯ = ⋯

instance HomologicalComplex.instHasHomologyUnopOfOpposite {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Preadditive V] (K : HomologicalComplex Vᵒᵖ c) (i : ι) [K.HasHomology i] : K.unop.HasHomology i

Equations

• ⋯ = ⋯

def HomologicalComplex.homologyOp {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Preadditive V] (K : HomologicalComplex V c) (i : ι) [K.HasHomology i] : K.op.homology i ≅ { unop := K.homology i }

If K is a homological complex, then the homology of K.op identifies to the opposite of the homology of K.

Equations

• K.homologyOp i = (K.sc i).homologyOpIso

def HomologicalComplex.homologyUnop {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Preadditive V] (K : HomologicalComplex Vᵒᵖ c) (i : ι) [K.HasHomology i] : K.unop.homology i ≅ (K.homology i).unop

If K is a homological complex in the opposite category, then the homology of K.unop identifies to the opposite of the homology of K.

Equations

• K.homologyUnop i = (K.unop.homologyOp i).unop

def HomologicalComplex.homology'OpDef {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Abelian V] (C : HomologicalComplex V c) (i : ι) : C.op.homology' i ≅ homology' (C.dFrom i).op (C.dTo i).op ⋯

Auxiliary tautological definition for homologyOp.

Equations

• C.homology'OpDef i = CategoryTheory.Iso.refl (C.op.homology' i)

def HomologicalComplex.homology'Op {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Abelian V] (C : HomologicalComplex V c) (i : ι) : C.op.homology' i ≅ { unop := C.homology' i }

Given a complex C of objects in V, the ith homology of its 'opposite' complex (with objects in Vᵒᵖ) is the opposite of the ith homology of C.

Equations

• C.homology'Op i = C.homology'OpDef i ≪≫ homology'Op (C.dTo i) (C.dFrom i) ⋯

def HomologicalComplex.homology'UnopDef {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Abelian V] (i : ι) (C : HomologicalComplex Vᵒᵖ c) : C.unop.homology' i ≅ homology' (C.dFrom i).unop (C.dTo i).unop ⋯

Auxiliary tautological definition for homologyUnop.

Equations

• HomologicalComplex.homology'UnopDef i C = CategoryTheory.Iso.refl (C.unop.homology' i)

def HomologicalComplex.homology'Unop {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Abelian V] (i : ι) (C : HomologicalComplex Vᵒᵖ c) : C.unop.homology' i ≅ (C.homology' i).unop

Given a complex C of objects in Vᵒᵖ, the ith homology of its 'opposite' complex (with objects in V) is the opposite of the ith homology of C.

Equations

• HomologicalComplex.homology'Unop i C = HomologicalComplex.homology'UnopDef i C ≪≫ homology'Unop (C.dTo i) (C.dFrom i) ⋯