Documentation

Mathlib.Algebra.Homology.Opposite

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

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 Opposite.op (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.

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    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.

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      @[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) :
      (HomologicalComplex.op X).d i j = (X.d j i).op

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

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        @[simp]

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

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          @[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) :
          (HomologicalComplex.unop X).d i j = (X.d j i).unop

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

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          • HomologicalComplex.unop X = { X := fun (i : ι) => (X.X i).unop, d := fun (i j : ι) => (X.d j i).unop, shape := , d_comp_d' := }
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            Sends a complex X with objects in Vᵒᵖ to the corresponding complex with objects in V.

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

              Auxiliary definition for opEquivalence.

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                @[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 (ComplexShape.symm c)} (f : X Y), (HomologicalComplex.opInverse V c).map f = { f := fun (i : ι) => (f.f i).unop, comm' := }.op

                Auxiliary definition for opEquivalence.

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                  Auxiliary definition for opEquivalence.

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                    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.

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                      @[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

                      Auxiliary definition for unopEquivalence.

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                        @[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 (ComplexShape.symm c)} (f : X Y), (HomologicalComplex.unopInverse V c).map f = { f := fun (i : ι) => (f.f i).op, comm' := }.op

                        Auxiliary definition for unopEquivalence.

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                          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.

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                            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.

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                              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.

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                                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.

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