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ComplexesCategories

Category of (co)chain complexes of an additive category

2022.09-02

7 September 2022

Kamal Saleh
Email: kamal.saleh@uni-siegen.de
Homepage: https://github.com/kamalsaleh/
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Contents

1 Complexes categories

1.1 Constructing chain and cochain categories

  1.1-1 IsChainOrCochainComplexCategory
  1.1-2 IsBoundedChainOrCochainComplexCategory
  1.1-3 IsChainComplexCategory
  1.1-4 IsBoundedChainComplexCategory
  1.1-5 IsCochainComplexCategory
  1.1-6 IsBoundedCochainComplexCategory
  1.1-7 ChainComplexCategory
  1.1-8 CochainComplexCategory
  1.1-9 UnderlyingCategory
  1.1-10 FullSubcategoryGeneratedByComplexesConcentratedInDegree
  1.1-11 AddIsNullHomotopic

2.1 Categories and filters

2.1-1 IsChainOrCochainComplex

2.2 Creating chain and cochain complexes

  2.2-1 ChainComplex
  2.2-2 ChainComplex
  2.2-3 ChainComplex
  2.2-4 StalkChainComplex
  2.2-5 ChainComplexWithInductiveSides
  2.2-6 ChainComplexWithInductiveSides
  2.2-7 CochainComplexWithInductiveSides
  2.2-8 ChainComplexWithInductiveNegativeSide
  2.2-9 ChainComplexWithInductivePositiveSide
  2.2-10 CochainComplexWithInductiveNegativeSide
  2.2-11 CochainComplexWithInductivePositiveSide

  2.3-1 Differentials
  2.3-2 Objects
  2.3-3 AsChainComplex
  2.3-4 AsCochainComplex
  2.3-5 IsExact
  2.3-6 IsContractable

2.5 Truncations

  2.5-1 GoodTruncationBelow
  2.5-2 GoodTruncationAbove
  2.5-3 GoodTruncationAbove
  2.5-4 GoodTruncationBelow
  2.5-5 BrutalTruncationBelow
  2.5-6 BrutalTruncationAbove
  2.5-7 BrutalTruncationAbove
  2.5-8 BrutalTruncationBelow

3 Complexes morphisms

3.1 Categories and filters

3.1-1 IsChainOrCochainMorphism

3.2 Creating chain and cochain morphisms

  3.2-1 ChainMorphism
  3.2-2 ChainMorphism
  3.2-3 ChainMorphism
  3.2-4 CochainMorphism
  3.2-5 CochainMorphism
  3.2-6 CochainMorphism
  3.2-7 StalkChainMorphism
  3.2-8 HomologyFunctorialAt

  3.3-1 Morphisms
  3.3-2 AsChainMorphism
  3.3-3 AsCochainMorphism
  3.3-4 MappingCone
  3.3-5 NaturalInjectionInMappingCone
  3.3-6 NaturalProjectionFromMappingCone
  3.3-7 MappingCylinder
  3.3-8 NaturalInjectionOfSourceInMappingCylinder
  3.3-9 NaturalInjectionOfRangeInMappingCylinder
  3.3-10 NaturalMorphismFromMappingCylinderInRange
  3.3-11 NaturalMorphismFromMappingCylinderInMappingCone
  3.3-12 HomotopyMorphisms

3.4-1 IsQuasiIsomorphism
3.4-2 IsNullHomotopic

  3.5-1 SetUpperBound
  3.5-2 SetLowerBound
  3.5-3 HasActiveUpperBound
  3.5-4 HasActiveLowerBound
  3.5-5 ActiveUpperBound
  3.5-6 ActiveLowerBound
  3.5-7 MorphismAt
  3.5-8 CyclesFunctorialAt
  3.5-9 \[\]
  3.5-10 IsQuasiIsomorphism
  3.5-11 MorphismsSupport
  3.5-12 Display
  3.5-13 IsWellDefined

4.1 Basic functors for complex categories.

  4.1-1 HomologyFunctor
  4.1-2 ShiftFunctor
  4.1-3 UnsignedShiftFunctor
  4.1-4 ChainToCochainComplexFunctor
  4.1-5 CochainToChainComplexFunctor
  4.1-6 ExtendFunctorToChainComplexCategories
  4.1-7 ExtendFunctorToCochainComplexCategories
  4.1-8 BrutalTruncationAboveFunctor
  4.1-9 BrutalTruncationBelowFunctor

4.2-1 GoodTruncationAboveFunctor

4.3 Natural transformations

4.3-1 ExtendNaturalTransformationToChainComplexCategories
4.3-2 ExtendNaturalTransformationToCochainComplexCategories

6.1 Definitions

6.2 Computing resolutions

  6.2-1 IsAbelianCategoryWithComputableEnoughProjectives
  6.2-2 IsAbelianCategoryWithComputableEnoughInjectives
  6.2-3 ProjectiveResolution
  6.2-4 ProjectiveResolution
  6.2-5 ProjectiveCochainResolution
  6.2-6 ProjectiveCochainResolution
  6.2-7 ProjectiveChainResolution
  6.2-8 ProjectiveChainResolution
  6.2-9 MorphismBetweenProjectiveResolutions
  6.2-10 MorphismBetweenProjectiveResolutions
  6.2-11 MorphismBetweenProjectiveCochainResolutions
  6.2-12 MorphismBetweenProjectiveCochainResolutions
  6.2-13 MorphismBetweenProjectiveChainResolutions
  6.2-14 MorphismBetweenProjectiveChainResolutions
  6.2-15 InjectiveResolution
  6.2-16 InjectiveResolution
  6.2-17 InjectiveCochainResolution
  6.2-18 InjectiveCochainResolution
  6.2-19 InjectiveChainResolution
  6.2-20 InjectiveChainResolution
  6.2-21 MorphismBetweenInjectiveResolutions
  6.2-22 MorphismBetweenInjectiveResolutions
  6.2-23 MorphismBetweenInjectiveCochainResolutions
  6.2-24 MorphismBetweenInjectiveCochainResolutions
  6.2-25 MorphismBetweenInjectiveChainResolutions
  6.2-26 MorphismBetweenInjectiveChainResolutions
  6.2-27 QuasiIsomorphismFromProjectiveResolution
  6.2-28 QuasiIsomorphismFromProjectiveResolution
  6.2-29 QuasiIsomorphismIntoInjectiveResolution
  6.2-30 QuasiIsomorphismIntoInjectiveResolution

6.3-1 MorphismFromHorseshoeResolution
6.3-2 HorseshoeResolution

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