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### 1 The bounded Z-graded closure of a category

#### 1.1 Constructors

 ‣ 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:

• PositivelyZGradedClosureCategory,

• NegativelyZGradedClosureCategory,

• FinitelyZGradedClosureCategory,

respectively.

 ‣ 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.

 ‣ 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

 ‣ IsZGradedClosureCategoryWithBounds( object ) ( filter )

Returns: true or false

The GAP category of Z-graded categories with bounds.

 ‣ IsCellInZGradedClosureCategoryWithBounds( object ) ( filter )

Returns: true or false

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

 ‣ IsObjectInZGradedClosureCategoryWithBounds( object ) ( filter )

Returns: true or false

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

 ‣ 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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