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TriangulatedCategories

Framework for triangulated categories

2024.09-03

14 September 2024

Kamal Saleh
Email: kamal.saleh@uni-siegen.de
Homepage: https://github.com/kamalsaleh
Address:
Department Mathematik
Universität Siegen
Walter-Flex-Straße 3
57072 Siegen
Germany

Contents

1 Category of triangles

1.1 Gap categories

  1.1-1 IsCategoryOfExactTriangles
  1.1-2 IsCategoryOfExactTrianglesObject
  1.1-3 IsCategoryOfExactTrianglesMorphism

1.2 Constructors

  1.2-1 CategoryOfExactTriangles
  1.2-2 UnderlyingCategory
  1.2-3 ExactTriangle
  1.2-4 DomainMorphism
  1.2-5 MorphismIntoConeObject
  1.2-6 MorphismFromConeObject
  1.2-7 ObjectAt
  1.2-8 \[\]
  1.2-9 MorphismAt
  1.2-10 \^
  1.2-11 StandardExactTriangle
  1.2-12 StandardExactTriangle
  1.2-13 IsStandardExactTriangle
  1.2-14 WitnessIsomorphismIntoStandardExactTriangle
  1.2-15 WitnessIsomorphismFromStandardExactTriangle
  1.2-16 MorphismOfExactTriangles
  1.2-17 MorphismAt
  1.2-18 \[\]
  1.2-19 MorphismBetweenConeObjects
  1.2-20 MorphismOfExactTriangles
  1.2-21 ExactTriangleByOctahedralAxiom
  1.2-22 ExactTriangleByOctahedralAxiom
  1.2-23 ExactTriangleByOctahedralAxiom
  1.2-24 ExactTriangleByOctahedralAxiom
  1.2-25 Rotation
  1.2-26 Rotation
  1.2-27 InverseRotation
  1.2-28 InverseRotation

2.1 Functors and natural transformations

  2.1-1 ShiftFunctor
  2.1-2 InverseShiftFunctor
  2.1-3 UnitOfShiftAdjunction
  2.1-4 InverseOfUnitOfShiftAdjunction
  2.1-5 CounitOfShiftAdjunction
  2.1-6 InverseOfCounitOfShiftAdjunction
  2.1-7 CommutativityNaturalTransformationWithShiftFunctor
  2.1-8 ExtendFunctorToCategoryOfTriangles
  2.1-9 RotationFunctor
  2.1-10 InverseRotationFunctor

3 Triangulated Categories

3.1 Categorical operations

  3.1-1 StandardConeObject
  3.1-2 MorphismIntoStandardConeObjectWithGivenStandardConeObject
  3.1-3 MorphismIntoStandardConeObject
  3.1-4 MorphismFromStandardConeObjectWithGivenObjects
  3.1-5 MorphismFromStandardConeObject
  3.1-6 ShiftOfObject
  3.1-7 ShiftOfMorphismWithGivenObjects
  3.1-8 ShiftOfMorphism
  3.1-9 Shift
  3.1-10 InverseShiftOfObject
  3.1-11 InverseShiftOfMorphismWithGivenObjects
  3.1-12 InverseShiftOfMorphism
  3.1-13 InverseShift
  3.1-14 UnitOfShiftAdjunctionWithGivenObject
  3.1-15 UnitOfShiftAdjunction
  3.1-16 InverseOfCounitOfShiftAdjunctionWithGivenObject
  3.1-17 InverseOfCounitOfShiftAdjunction
  3.1-18 InverseOfUnitOfShiftAdjunctionWithGivenObject
  3.1-19 InverseOfUnitOfShiftAdjunction
  3.1-20 CounitOfShiftAdjunctionWithGivenObject
  3.1-21 CounitOfShiftAdjunction
  3.1-22 MorphismBetweenStandardConeObjectsWithGivenObjects
  3.1-23 MorphismBetweenStandardConeObjects
  3.1-24 ConeObjectByOctahedralAxiom
  3.1-25 DomainMorphismByOctahedralAxiomWithGivenObjects
  3.1-26 MorphismIntoConeObjectByOctahedralAxiomWithGivenObjects
  3.1-27 MorphismFromConeObjectByOctahedralAxiomWithGivenObjects
  3.1-28 WitnessIsomorphismIntoStandardConeObjectByOctahedralAxiomWithGivenObjects
  3.1-29 WitnessIsomorphismIntoStandardConeObjectByOctahedralAxiom
  3.1-30 WitnessIsomorphismFromStandardConeObjectByOctahedralAxiomWithGivenObjects
  3.1-31 WitnessIsomorphismFromStandardConeObjectByOctahedralAxiom
  3.1-32 ConeObjectByRotationAxiom
  3.1-33 DomainMorphismByRotationAxiom
  3.1-34 MorphismIntoConeObjectByRotationAxiom
  3.1-35 MorphismFromConeObjectByRotationAxiom
  3.1-36 WitnessIsomorphismIntoStandardConeObjectByRotationAxiom
  3.1-37 WitnessIsomorphismIntoStandardConeObjectByRotationAxiomWithGivenObjects
  3.1-38 WitnessIsomorphismFromStandardConeObjectByRotationAxiom
  3.1-39 WitnessIsomorphismFromStandardConeObjectByRotationAxiomWithGivenObjects
  3.1-40 ConeObjectByInverseRotationAxiom
  3.1-41 DomainMorphismByInverseRotationAxiom
  3.1-42 MorphismIntoConeObjectByInverseRotationAxiom
  3.1-43 MorphismFromConeObjectByInverseRotationAxiom
  3.1-44 WitnessIsomorphismIntoStandardConeObjectByInverseRotationAxiom
  3.1-45 WitnessIsomorphismIntoStandardConeObjectByInverseRotationAxiomWithGivenObjects
  3.1-46 WitnessIsomorphismFromStandardConeObjectByInverseRotationAxiom
  3.1-47 WitnessIsomorphismFromStandardConeObjectByInverseRotationAxiomWithGivenObjects
  3.1-48 ShiftExpandingIsomorphismWithGivenObjects
  3.1-49 ShiftExpandingIsomorphism
  3.1-50 ShiftFactoringIsomorphismWithGivenObjects
  3.1-51 ShiftFactoringIsomorphism
  3.1-52 InverseShiftExpandingIsomorphismWithGivenObjects
  3.1-53 InverseShiftExpandingIsomorphism
  3.1-54 InverseShiftFactoringIsomorphismWithGivenObjects
  3.1-55 InverseShiftFactoringIsomorphism

4 operations for triangulated categories

4.1 Add-methods

  4.1-1 AddCounitOfShiftAdjunctionWithGivenObject
  4.1-2 AddDomainMorphismByOctahedralAxiomWithGivenObjects
  4.1-3 AddInverseOfCounitOfShiftAdjunctionWithGivenObject
  4.1-4 AddInverseOfUnitOfShiftAdjunctionWithGivenObject
  4.1-5 AddInverseShiftExpandingIsomorphismWithGivenObjects
  4.1-6 AddInverseShiftFactoringIsomorphismWithGivenObjects
  4.1-7 AddInverseShiftOfMorphismWithGivenObjects
  4.1-8 AddInverseShiftOfObject
  4.1-9 AddIsExactTriangle
  4.1-10 AddMorphismBetweenStandardConeObjectsWithGivenObjects
  4.1-11 AddMorphismFromConeObjectByOctahedralAxiomWithGivenObjects
  4.1-12 AddMorphismFromStandardConeObject
  4.1-13 AddMorphismFromStandardConeObjectWithGivenObjects
  4.1-14 AddMorphismIntoConeObjectByOctahedralAxiomWithGivenObjects
  4.1-15 AddMorphismIntoStandardConeObject
  4.1-16 AddMorphismIntoStandardConeObjectWithGivenStandardConeObject
  4.1-17 AddShiftExpandingIsomorphismWithGivenObjects
  4.1-18 AddShiftFactoringIsomorphismWithGivenObjects
  4.1-19 AddShiftOfMorphismByInteger
  4.1-20 AddShiftOfMorphismByIntegerWithGivenObjects
  4.1-21 AddShiftOfMorphismWithGivenObjects
  4.1-22 AddShiftOfObject
  4.1-23 AddShiftOfObjectByInteger
  4.1-24 AddStandardConeObject
  4.1-25 AddUnitOfShiftAdjunctionWithGivenObject
  4.1-26 AddWitnessIsomorphismFromStandardConeObject
  4.1-27 AddWitnessIsomorphismFromStandardConeObjectByInverseRotationAxiomWithGivenObjects
  4.1-28 AddWitnessIsomorphismFromStandardConeObjectByOctahedralAxiomWithGivenObjects
  4.1-29 AddWitnessIsomorphismFromStandardConeObjectByRotationAxiomWithGivenObjects
  4.1-30 AddWitnessIsomorphismIntoStandardConeObject
  4.1-31 AddWitnessIsomorphismIntoStandardConeObjectByInverseRotationAxiomWithGivenObjects
  4.1-32 AddWitnessIsomorphismIntoStandardConeObjectByOctahedralAxiomWithGivenObjects
  4.1-33 AddWitnessIsomorphismIntoStandardConeObjectByRotationAxiomWithGivenObjects

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