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1 The bounded Z-graded closure of a category
 1.1 Constructors
 1.2 Operations
 1.3 GAP Categories
 1.4 Attributes

1 The bounded Z-graded closure of a category

1.1 Constructors

1.1-1 ZGradedClosureCategoryWithBounds
‣ ZGradedClosureCategoryWithBounds( C, str )( operation )

Returns: a CAP category

Construct the \mathbb{Z}-graded closure of the category C with bounds. The string str can be either "lower", "upper", or "both", implementing the three constructors:

respectively.

1.1-2 ObjectInZGradedClosureCategoryWithBounds
‣ ObjectInZGradedClosureCategoryWithBounds( ZC, L )( operation )
‣ ObjectInZGradedClosureCategoryWithBounds( ZC, f, lower_bound, upper_bound )( operation )
‣ ObjectInZGradedClosureCategoryWithBounds( ZC, M, degree )( operation )
‣ ObjectInZGradedClosureCategoryWithBounds( ZC, M )( operation )
‣ ObjectInZGradedClosureCategoryWithBounds( L )( operation )

Returns: a CAP object

Construct an object in the bounded \mathbb{Z}-graded category ZC using the Z-function L.

1.1-3 MorphismInZGradedClosureCategoryWithBounds
‣ MorphismInZGradedClosureCategoryWithBounds( S, L, T )( operation )
‣ MorphismInZGradedClosureCategoryWithBounds( S, f, T )( operation )
‣ MorphismInZGradedClosureCategoryWithBounds( S, phi, degree, T )( operation )

Returns: a CAP morphism

Construct a morphism in a bounded \mathbb{Z}-graded category.

1.1-4 ComponentInclusionMorphism
‣ ComponentInclusionMorphism( M, chi, degree, i )( operation )
‣ ComponentInclusionMorphism( M, chi, i )( operation )
‣ ComponentInclusionMorphism( M, degree )( operation )

Returns: a CAP morphism

1.1-5 DiagonalEmbeddingWithGivenDegrees
‣ DiagonalEmbeddingWithGivenDegrees( M, degrees )( operation )

1.1-6 DiagonalEmbedding
‣ DiagonalEmbedding( M )( operation )
‣ DiagonalEmbedding( S, M )( operation )

1.2 Operations

1.2-1 ActiveLowerBound
‣ ActiveLowerBound( c )( operation )

Returns: an integer or infinity

The active lower bound of the cell (=object or morphism) c.

1.2-2 SetActiveLowerBound
‣ SetActiveLowerBound( c, lower_bound )( operation )

Returns: an integer or infinity

Set the active lower bound of the cell (=object or morphism) c to lower_bound if it is greater than the active lower bound, and return it.

1.2-3 ActiveUpperBound
‣ ActiveUpperBound( c )( operation )

Returns: an integer or infinity

The active upper bound of the cell (=object or morphism) c.

1.2-4 SetActiveUpperBound
‣ SetActiveUpperBound( c, upper_bound )( operation )

Returns: an integer or infinity

Set the active upper bound of the cell (=object or morphism) c to upper_bound if it is less than the active upper bound, and return it.

1.2-5 TensorProductIndices
‣ TensorProductIndices( A, B )( operation )

Returns: a function

Returns the function f: n |-> [ ActiveLowerBound( A ) .. n - ActiveLowerBound( B ) ] over which to run the tensor product summation A[i] \otimes B[n - i] (i \in f(n)) for (A \otimes B)[n].

1.2-6 TensorProductIndices
‣ TensorProductIndices( A, B )( operation )

Returns: a pair of functions

Returns two functions over which to run the tensor product summation for (A \otimes (B \otimes C))[n] resp. for ((A \otimes B) \otimes C)[n].

1.2-7 []
‣ []( c, n )( operation )

Returns: a CAP category

The i-th object of the infinite list underlying the cell (=object or morphism) c.

1.2-8 CertainDegreePart
‣ CertainDegreePart( c, n )( operation )

Returns: a CAP category

The i-th object of the infinite list(s) underlying the cell resp. list c.

1.2-9 Sublist
‣ Sublist( c, L )( operation )

Returns: a CAP category

The L-th sublist of the infinite list underlying the cell (=object or morphism) c.

1.3 GAP Categories

1.3-1 IsZGradedClosureCategoryWithBounds
‣ IsZGradedClosureCategoryWithBounds( object )( filter )

Returns: true or false

The GAP category of Z-graded categories with bounds.

1.3-2 IsCellInZGradedClosureCategoryWithBounds
‣ IsCellInZGradedClosureCategoryWithBounds( object )( filter )

Returns: true or false

The GAP category of cells in a Z-graded category with bounds.

1.3-3 IsObjectInZGradedClosureCategoryWithBounds
‣ IsObjectInZGradedClosureCategoryWithBounds( object )( filter )

Returns: true or false

The GAP category of objects in a Z-graded category with bounds.

1.3-4 IsMorphismInZGradedClosureCategoryWithBounds
‣ IsMorphismInZGradedClosureCategoryWithBounds( morphism )( filter )

Returns: true or false

The GAP category of morphisms in a Z-graded category with bounds.

1.4 Attributes

1.4-1 UnderlyingCategory
‣ UnderlyingCategory( UC )( attribute )

Return the category C underlying the Z-graded category with bounds category ZC := BoundedZGradedCategory( C )).

1.4-2 UnderlyingZFunctionAndBounds
‣ UnderlyingZFunctionAndBounds( obj )( attribute )

Returns: a pair including a Z-function and a pair of integers

The \mathbb{Z}-function underlying the object obj.

1.4-3 UnderlyingZFunction
‣ UnderlyingZFunction( mor )( attribute )

Returns: a Z-function

The \mathbb{Z}-function underlying the morphism mor.

1.4-4 NonZeroParts
‣ NonZeroParts( object )( attribute )

Returns: a list

The support of the object c.

1.4-5 NonZeroDegrees
‣ NonZeroDegrees( object )( attribute )

Returns: a list

The list of degrees of the support of the object c.

1.4-6 NonZeroDegreesHull
‣ NonZeroDegreesHull( object )( attribute )

Returns: a list

A list of integers containing the list of degrees of the support of the object c.

1.4-7 NonZeroPartsWithDegrees
‣ NonZeroPartsWithDegrees( object )( attribute )

Returns: a list

1.4-8 SupportWithDegrees
‣ SupportWithDegrees( object )( attribute )

Returns: a list

1.4-9 SupportWithDegreesWithGivenDegrees
‣ SupportWithDegreesWithGivenDegrees( object, L )( operation )

Like SupportWithDegrees but only considers the degrees in the given list L.

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