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

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

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

Equations
  • X.op = { X := fun (i : ι) => Opposite.op (X.X i), d := fun (i j : ι) => (X.d j i).op, shape := , d_comp_d' := }
Instances For
    @[simp]
    theorem HomologicalComplex.opSymm_d {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Limits.HasZeroMorphisms V] (X : HomologicalComplex V c.symm) (i : ι) (j : ι) :
    X.opSymm.d i j = (X.d j i).op
    @[simp]
    theorem HomologicalComplex.opSymm_X {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Limits.HasZeroMorphisms V] (X : HomologicalComplex V c.symm) (i : ι) :
    X.opSymm.X i = Opposite.op (X.X i)

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

    Equations
    • X.opSymm = { X := fun (i : ι) => Opposite.op (X.X i), d := fun (i j : ι) => (X.d j i).op, shape := , d_comp_d' := }
    Instances For
      @[simp]
      theorem HomologicalComplex.unop_d {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Limits.HasZeroMorphisms V] (X : HomologicalComplex Vᵒᵖ c) (i : ι) (j : ι) :
      X.unop.d i j = (X.d j i).unop

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

      Equations
      • X.unop = { X := fun (i : ι) => Opposite.unop (X.X i), d := fun (i j : ι) => (X.d j i).unop, shape := , d_comp_d' := }
      Instances For
        @[simp]
        theorem HomologicalComplex.unopSymm_d {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Limits.HasZeroMorphisms V] (X : HomologicalComplex Vᵒᵖ c.symm) (i : ι) (j : ι) :
        X.unopSymm.d i j = (X.d j i).unop
        @[simp]

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

        Equations
        • X.unopSymm = { X := fun (i : ι) => Opposite.unop (X.X i), d := fun (i j : ι) => (X.d j i).unop, shape := , d_comp_d' := }
        Instances For
          @[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.Limits.HasZeroMorphisms 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.

          Equations
          • One or more equations did not get rendered due to their size.
          Instances For
            @[simp]
            theorem HomologicalComplex.opInverse_map {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Limits.HasZeroMorphisms 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' := }

            Auxiliary definition for opEquivalence.

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

              Auxiliary definition for opEquivalence.

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

                Auxiliary definition for opEquivalence.

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

                  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.
                  Instances For
                    @[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.Limits.HasZeroMorphisms 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.

                    Equations
                    • One or more equations did not get rendered due to their size.
                    Instances For
                      @[simp]
                      theorem HomologicalComplex.unopInverse_map {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Limits.HasZeroMorphisms 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' := }

                      Auxiliary definition for unopEquivalence.

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

                        Auxiliary definition for unopEquivalence.

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

                          Auxiliary definition for unopEquivalence.

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

                            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.
                            Instances For
                              instance HomologicalComplex.instHasHomologyOppositeOp {ι : Type u_1} (V : Type u_2) [CategoryTheory.Category.{u_3, u_2} V] (c : ComplexShape ι) [CategoryTheory.Limits.HasZeroMorphisms 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.Limits.HasZeroMorphisms 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.Limits.HasZeroMorphisms V] (K : HomologicalComplex V c) (i : ι) [K.HasHomology i] :
                              K.op.homology i Opposite.op (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
                              Instances For
                                def HomologicalComplex.homologyUnop {ι : Type u_1} {V : Type u_2} [CategoryTheory.Category.{u_3, u_2} V] {c : ComplexShape ι} [CategoryTheory.Limits.HasZeroMorphisms V] (K : HomologicalComplex Vᵒᵖ c) (i : ι) [K.HasHomology i] :
                                K.unop.homology i Opposite.unop (K.homology i)

                                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
                                Instances For