The homology of single complexes #
The main definition in this file is HomologicalComplex.homologyFunctorSingleIso
which is a natural isomorphism single C c j ⋙ homologyFunctor C c j ≅ 𝟭 C
.
Equations
- ⋯ = ⋯
The canonical isomorphism ((single C c j).obj A).cycles j ≅ A
Equations
- HomologicalComplex.singleObjCyclesSelfIso c j A = ((HomologicalComplex.single C c j).obj A).iCyclesIso j (c.next j) ⋯ ⋯ ≪≫ HomologicalComplex.singleObjXSelf c j A
Instances For
The canonical isomorphism ((single C c j).obj A).opcycles j ≅ A
Equations
- HomologicalComplex.singleObjOpcyclesSelfIso c j A = (HomologicalComplex.singleObjXSelf c j A).symm ≪≫ ((HomologicalComplex.single C c j).obj A).pOpcyclesIso (c.prev j) j ⋯ ⋯
Instances For
The canonical isomorphism ((single C c j).obj A).homology j ≅ A
Equations
- HomologicalComplex.singleObjHomologySelfIso c j A = (((HomologicalComplex.single C c j).obj A).isoHomologyπ (c.prev j) j ⋯ ⋯).symm ≪≫ HomologicalComplex.singleObjCyclesSelfIso c j A
Instances For
The computation of the homology of single complexes, as a natural isomorphism
single C c j ⋙ homologyFunctor C c j ≅ 𝟭 C
.
Equations
- HomologicalComplex.homologyFunctorSingleIso C c j = CategoryTheory.NatIso.ofComponents (fun (A : C) => HomologicalComplex.singleObjHomologySelfIso c j A) ⋯
Instances For
Sending objects to chain complexes supported at 0
then taking 0
-th homology
is the same as doing nothing.
Equations
- ChainComplex.homology'Functor0Single₀ C = CategoryTheory.NatIso.ofComponents (fun (X : C) => homology'.congr ⋯ ⋯ ⋯ ⋯ ≪≫ homology'ZeroZero) ⋯
Instances For
Sending objects to chain complexes supported at 0
then taking (n+1)
-st homology
is the same as the zero functor.
Equations
- ChainComplex.homology'FunctorSuccSingle₀ C n = CategoryTheory.NatIso.ofComponents (fun (X : C) => homology'.congr ⋯ ⋯ ⋯ ⋯ ≪≫ homology'ZeroZero ≪≫ ⋯.isoZero.symm) ⋯
Instances For
Sending objects to cochain complexes supported at 0
then taking 0
-th homology
is the same as doing nothing.
Equations
- CochainComplex.homologyFunctor0Single₀ C = CategoryTheory.NatIso.ofComponents (fun (X : C) => homology'.congr ⋯ ⋯ ⋯ ⋯ ≪≫ homology'ZeroZero) ⋯
Instances For
Sending objects to cochain complexes supported at 0
then taking (n+1)
-st homology
is the same as the zero functor.
Equations
- CochainComplex.homology'FunctorSuccSingle₀ C n = CategoryTheory.NatIso.ofComponents (fun (X : C) => homology'.congr ⋯ ⋯ ⋯ ⋯ ≪≫ homology'ZeroZero ≪≫ ⋯.isoZero.symm) ⋯