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GradedCategories

Graded closures of categories

2025.07-01

16 July 2025

Mohamed Barakat
Email: mohamed.barakat@uni-siegen.de
Homepage: https://mohamed-barakat.github.io/
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Fabian Zickgraf
Email: fabian.zickgraf@uni-siegen.de
Homepage: https://algebra.mathematik.uni-siegen.de/zickgraf
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Contents

1 The bounded Z-graded closure of a category

1.1 Constructors

  1.1-1 ZGradedClosureCategoryWithBounds
  1.1-2 ObjectInZGradedClosureCategoryWithBounds
  1.1-3 MorphismInZGradedClosureCategoryWithBounds
  1.1-4 ComponentInclusionMorphism
  1.1-5 DiagonalEmbeddingWithGivenDegrees
  1.1-6 DiagonalEmbedding

  1.2-1 ActiveLowerBound
  1.2-2 SetActiveLowerBound
  1.2-3 ActiveUpperBound
  1.2-4 SetActiveUpperBound
  1.2-5 TensorProductIndices
  1.2-6 TensorProductIndices
  1.2-7 []
  1.2-8 CertainDegreePart
  1.2-9 Sublist

1.3 GAP Categories

  1.3-1 IsZGradedClosureCategoryWithBounds
  1.3-2 IsCellInZGradedClosureCategoryWithBounds
  1.3-3 IsObjectInZGradedClosureCategoryWithBounds
  1.3-4 IsMorphismInZGradedClosureCategoryWithBounds

  1.4-1 UnderlyingCategory
  1.4-2 UnderlyingZFunctionAndBounds
  1.4-3 UnderlyingZFunction
  1.4-4 NonZeroParts
  1.4-5 NonZeroDegrees
  1.4-6 NonZeroDegreesHull
  1.4-7 NonZeroPartsWithDegrees
  1.4-8 SupportWithDegrees
  1.4-9 SupportWithDegreesWithGivenDegrees

2 The positively Z-graded closure of a category

2.1 Constructors

2.1-1 PositivelyZGradedClosureCategory

2.2 GAP Categories

  2.2-1 IsPositivelyZGradedClosureCategory
  2.2-2 IsCellInPositivelyZGradedClosureCategory
  2.2-3 IsObjectInPositivelyZGradedClosureCategory
  2.2-4 IsMorphismInPositivelyZGradedClosureCategory

3 The negatively Z-graded closure of a category

3.1 Constructors

3.1-1 NegativelyZGradedClosureCategory

3.2 GAP Categories

  3.2-1 IsNegativelyZGradedClosureCategory
  3.2-2 IsCellInNegativelyZGradedClosureCategory
  3.2-3 IsObjectInNegativelyZGradedClosureCategory
  3.2-4 IsMorphismInNegativelyZGradedClosureCategory

4 The finitely Z-graded closure of a category

4.1 Constructors

4.1-1 FinitelyZGradedClosureCategory

4.2 GAP Categories

  4.2-1 IsFinitelyZGradedClosureCategory
  4.2-2 IsCellInFinitelyZGradedClosureCategory
  4.2-3 IsObjectInFinitelyZGradedClosureCategory
  4.2-4 IsMorphismInFinitelyZGradedClosureCategory

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