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TriangulatedCategories

Framework for triangulated categories

2022.10-01

28 October 2022

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 IsCapCategoryOfExactTriangles
  1.1-2 IsCapExactTriangle
  1.1-3 IsCapExactTrianglesMorphism

1.2 Constructors

  1.2-1 CategoryOfExactTriangles
  1.2-2 UnderlyingCategory
  1.2-3 ExactTriangle
  1.2-4 DomainMorphism
  1.2-5 MorphismToConeObject
  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 WitnessIsomorphismOntoStandardExactTriangle
  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 Unit
  2.1-4 InverseOfUnit
  2.1-5 Counit
  2.1-6 InverseOfCounit
  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 AddStandardConeObject
  3.1-3 MorphismToStandardConeObjectWithGivenStandardConeObject
  3.1-4 AddMorphismToStandardConeObjectWithGivenStandardConeObject
  3.1-5 MorphismToStandardConeObject
  3.1-6 AddMorphismToStandardConeObject
  3.1-7 MorphismFromStandardConeObjectWithGivenStandardConeObject
  3.1-8 AddMorphismFromStandardConeObjectWithGivenStandardConeObject
  3.1-9 MorphismFromStandardConeObject
  3.1-10 AddMorphismFromStandardConeObject
  3.1-11 ShiftOnObject
  3.1-12 AddShiftOnObject
  3.1-13 ShiftOnMorphismWithGivenObjects
  3.1-14 AddShiftOnMorphismWithGivenObjects
  3.1-15 ShiftOnMorphism
  3.1-16 Shift
  3.1-17 InverseShiftOnObject
  3.1-18 AddInverseShiftOnObject
  3.1-19 InverseShiftOnMorphismWithGivenObjects
  3.1-20 AddInverseShiftOnMorphismWithGivenObjects
  3.1-21 InverseShiftOnMorphism
  3.1-22 InverseShift
  3.1-23 UnitIsomorphismWithGivenObject
  3.1-24 AddUnitIsomorphismWithGivenObject
  3.1-25 UnitIsomorphism
  3.1-26 AddUnitIsomorphism
  3.1-27 InverseOfCounitIsomorphismWithGivenObject
  3.1-28 AddInverseOfCounitIsomorphismWithGivenObject
  3.1-29 InverseOfCounitIsomorphism
  3.1-30 InverseOfUnitIsomorphismWithGivenObject
  3.1-31 AddInverseOfUnitIsomorphismWithGivenObject
  3.1-32 InverseOfUnitIsomorphism
  3.1-33 CounitIsomorphismWithGivenObject
  3.1-34 AddCounitIsomorphismWithGivenObject
  3.1-35 CounitIsomorphism
  3.1-36 MorphismBetweenStandardConeObjectsWithGivenObjects
  3.1-37 AddMorphismBetweenStandardConeObjectsWithGivenObjects
  3.1-38 MorphismBetweenStandardConeObjects
  3.1-39 ConeObjectByOctahedralAxiom
  3.1-40 DomainMorphismByOctahedralAxiomWithGivenObjects
  3.1-41 AddDomainMorphismByOctahedralAxiomWithGivenObjects
  3.1-42 MorphismToConeObjectByOctahedralAxiomWithGivenObjects
  3.1-43 AddMorphismToConeObjectByOctahedralAxiomWithGivenObjects
  3.1-44 MorphismFromConeObjectByOctahedralAxiomWithGivenObjects
  3.1-45 AddMorphismFromConeObjectByOctahedralAxiomWithGivenObjects
  3.1-46 WitnessIsomorphismOntoStandardConeObjectByOctahedralAxiomWithGivenObjects
  3.1-47 AddWitnessIsomorphismOntoStandardConeObjectByOctahedralAxiomWithGivenObjects
  3.1-48 WitnessIsomorphismOntoStandardConeObjectByOctahedralAxiom
  3.1-49 WitnessIsomorphismFromStandardConeObjectByOctahedralAxiomWithGivenObjects
  3.1-50 AddWitnessIsomorphismFromStandardConeObjectByOctahedralAxiomWithGivenObjects
  3.1-51 WitnessIsomorphismFromStandardConeObjectByOctahedralAxiom
  3.1-52 ConeObjectByRotationAxiom
  3.1-53 DomainMorphismByRotationAxiom
  3.1-54 MorphismToConeObjectByRotationAxiom
  3.1-55 MorphismFromConeObjectByRotationAxiom
  3.1-56 WitnessIsomorphismOntoStandardConeObjectByRotationAxiom
  3.1-57 WitnessIsomorphismOntoStandardConeObjectByRotationAxiomWithGivenObjects
  3.1-58 AddWitnessIsomorphismOntoStandardConeObjectByRotationAxiomWithGivenObjects
  3.1-59 WitnessIsomorphismFromStandardConeObjectByRotationAxiom
  3.1-60 WitnessIsomorphismFromStandardConeObjectByRotationAxiomWithGivenObjects
  3.1-61 AddWitnessIsomorphismFromStandardConeObjectByRotationAxiomWithGivenObjects
  3.1-62 ConeObjectByInverseRotationAxiom
  3.1-63 DomainMorphismByInverseRotationAxiom
  3.1-64 MorphismToConeObjectByInverseRotationAxiom
  3.1-65 MorphismFromConeObjectByInverseRotationAxiom
  3.1-66 WitnessIsomorphismOntoStandardConeObjectByInverseRotationAxiom
  3.1-67 WitnessIsomorphismOntoStandardConeObjectByInverseRotationAxiomWithGivenObjects
  3.1-68 AddWitnessIsomorphismOntoStandardConeObjectByInverseRotationAxiomWithGivenObjects
  3.1-69 WitnessIsomorphismFromStandardConeObjectByInverseRotationAxiom
  3.1-70 WitnessIsomorphismFromStandardConeObjectByInverseRotationAxiomWithGivenObjects
  3.1-71 AddWitnessIsomorphismFromStandardConeObjectByInverseRotationAxiomWithGivenObjects
  3.1-72 ShiftExpandingIsomorphismWithGivenObjects
  3.1-73 AddShiftExpandingIsomorphismWithGivenObjects
  3.1-74 ShiftExpandingIsomorphism
  3.1-75 ShiftFactoringIsomorphismWithGivenObjects
  3.1-76 AddShiftFactoringIsomorphismWithGivenObjects
  3.1-77 ShiftFactoringIsomorphism
  3.1-78 InverseShiftExpandingIsomorphismWithGivenObjects
  3.1-79 AddInverseShiftExpandingIsomorphismWithGivenObjects
  3.1-80 InverseShiftExpandingIsomorphism
  3.1-81 InverseShiftFactoringIsomorphismWithGivenObjects
  3.1-82 AddInverseShiftFactoringIsomorphismWithGivenObjects
  3.1-83 InverseShiftFactoringIsomorphism

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