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2 Additive closure

2.1 GAP Categories

2.1-1 IsAdditiveClosureCategory

Returns: true or false

The GAP category of additive closures of Ab-categories.

2.1-2 IsAdditiveClosureObject

Returns: true or false

The GAP category of objects in additive closures of Ab-categories.

2.1-3 IsAdditiveClosureMorphism

Returns: true or false

The GAP category of morphisms in additive closures of Ab-categories.

2.2 Constructors

2.2-1 AdditiveClosure

Returns: the category \(C^\oplus\)

The argument is an Ab-category \(C\). The output is its additive closure \(C^\oplus\).

2.2-2 ADDITIVE_CLOSURE

Returns: the category \(C^\oplus\)

Same as AdditiveClosure (2.2-1), but as an operation instead of an attribute.

2.2-3 AdditiveClosureObject

Returns: an object in \(C^\oplus\)

The argument is a list of objects \(L=[A_1,\dots,A_n]\) in an Ab-category \(C\). The output is the formal direct sum \(A_1\oplus\dots\oplus A_n\) in the additive closure \(C^\oplus\).

2.2-4 AsAdditiveClosureObject

Returns: an object in \(C^\oplus\)

The argument is an object \(A\) in an Ab-category \(C\). The output is the image of \(A\) under the inclusion functor \(\iota:C\to C^\oplus\).

2.2-5 AdditiveClosureMorphism

Returns: a morphism in \(\mathrm{Hom}_{C^\oplus}(A,B)\)

The arguments are formal direct sums \(A=A_1\oplus\dots\oplus A_m\), \(B=B_1\oplus\dots\oplus B_n\) in some additive category \(C^\oplus\) and an \(m\times n\) matrix \(M :=(\alpha_{ij}:A_i\to B_j)_{ij}\) for \(i=1,\dots,m,j=1,\dots,n\). The output is the formal morphism between \(A\) and \(B\) that is defined by \(M\).

2.2-6 AsAdditiveClosureMorphism

Returns: a morphism in \(C^\oplus\)

The argument is a morphism \(\alpha\) in an Ab-category \(C\). The output is the image of \(\alpha\) under the inclusion functor \(\iota:C\to C^\oplus\).

2.2-7 InclusionFunctorInAdditiveClosure

Returns: a functor \(C\to C^\oplus\)

The argument is an Ab-category \(C\). The output is the inclusion functor \(\iota:C\to C^\oplus\).

2.2-8 ExtendFunctorToAdditiveClosures

Returns: a functor \(C^\oplus \to D^\oplus\)

The argument is a functor \(F:C\to D\), and the output is the extension functor \(F^\oplus:C^\oplus \to D^\oplus\).

2.2-9 ExtendFunctorWithAdditiveRangeToFunctorFromAdditiveClosureOfSource

Returns: a functor \(C^\oplus \to D\)

The argument is a functor \(F:C\to D\), where \(D\) is an additive category. The output is the extension functor \(F^\oplus:C^\oplus \to D\).

2.2-10 ExtendFunctorToAdditiveClosureOfSource

Returns: a functor \(C^\oplus \to D^\oplus\) or \(C^\oplus \to D\)

The argument is a functor \(F:C\to D\). If \(D\) is not known to be an additive category, then return ExtendFunctorToAdditiveClosures(F), otherwise return ExtendFunctorWithAdditiveRangeToFunctorFromAdditiveClosureOfSource(F).

2.2-11 ExtendNaturalTransformationToAdditiveClosureOfSource

Returns: a natural transformation from \(F^\oplus\) to \(G^\oplus\)

The argument is a natural transformation \(\eta:(F:C\to D)\Rightarrow (G:C\to D)\) where \(D\) is an additive category. The ouput is the extension natural transformation \(\eta^\oplus:(F^\oplus:C^\oplus\to D)\to(G^\oplus:C^\oplus\to D)\).

2.3 Attributes

2.3-1 UnderlyingCategory

Returns: the category \(C\)

The argument is some additive closure category \(A:=C^\oplus\). The output is \(C\).

2.3-2 ObjectList

Returns: a list of the objects in \(C\)

The argument is a formal direct sum \(A:=A_1\oplus\dots\oplus A_m\) in some additive closure category \(C^\oplus\). The output is the list \([A_1,\dots,A_m]\).

2.3-3 MorphismMatrix

Returns: a list of lists the morphisms in \(C\)

The argument is a morphism \(\alpha:A\to B\) between formal direct sums in some additive closure category \(C^\oplus\). The output is the defining matrix of \(\alpha\).

2.3-4 NumberRows

Returns: a non-negative integer

The argument is a morphism \(\alpha:A\to B\) between formal direct sums. The output is the number of summands of the the source.

2.3-5 NumberColumns

Returns: a non-negative integer

The argument is a morphism \(\alpha:A\to B\) between formal direct sums. The output is the number of summands of the the range.

2.4 Operators

2.4-1 []

Returns: an object in \(C\)

The arguments are a formal direct sum \(A\) in some additive category \(C^\oplus\) and an integers \(i\). The output is the \(i\)'th entry in ObjectList(\(A\)).

2.4-2 [

Returns: a morphism \(C\)

The arguments are a morphism \(\alpha:A\to B\) between formal direct sums in some additive category \(C^\oplus\) and two integers \(i,j\). The output is the \((i,j)\)'th entry in MorphismMatrix(\(\alpha\)).

2.4-3 /

The input is either a list of objects or list of lists of morphisms. The method delegates to either AdditiveClosureObject or AdditiveClosureMorphism.

2.4-4 /

This is a convenience method for AsAdditiveClosureObject and AsAdditiveClosureMorphism.

2.5 Random methods for additive closure category

2.5-1 RandomObjectByList

Returns: an object in C

The arguments are an additive closure category C of a category U and a list L whose first entry is a non-empty list of non-negative integers and second entry is a list. The output is an object in C which is a formal direct sum of Random(L[1]) objects in U each computed via RandomObjectByList(U,L[2]).

2.5-2 RandomObjectByInteger

Returns: an object in C

The arguments are an additive closure category C of a category U and a non-negative integer n. The output is an object in C which is a formal direct sum of at most n objects in U each computed via RandomObjectByInteger(U,n).

2.5-3 RandomMorphismWithFixedSourceAndRangeByList

Returns: a morphism in C

The arguments are two objects S, R and a list L. The output is a morphism from S to R whose matrix entry at index \(i,j\) is computed via RandomMorphismWithFixedSourceAndRangeByList(S[\(i\)],R[\(j\)],L).

2.5-4 RandomMorphismWithFixedSourceAndRangeByInteger

Returns: a morphism in C

The arguments are two objects S, R and an integer n. The output is a morphism from S to R whose matrix entry at index \(i,j\) is computed via RandomMorphismWithFixedSourceAndRangeByInteger(S[\(i\)],R[\(j\)],n).

2.5-5 RandomMorphismWithFixedSourceByList

Returns: a morphism in C

The arguments are an object S in an additive closure category C and a list L consisting of two lists. The output is RandomMorphismWithFixedSourceAndRangeByList(S,\(R\),L[2]) where \(R\) is computed via RandomObjectByList(C,L[1]).

2.5-6 RandomMorphismWithFixedSourceByInteger

Returns: a morphism in C

The arguments are an object S in an additive closure category C and an integer n. The output is RandomMorphismWithFixedSourceAndRangeByInteger(S,\(R\),\(1\)+Log2Int(n)) where \(R\) is computed via RandomObjectByInteger(C,n).

2.5-7 RandomMorphismWithFixedRangeByList

Returns: a morphism in a category of rows

The arguments are an object R in an additive closure category C and a list L consisting of two lists. The output is RandomMorphismWithFixedSourceAndRangeByList(S,\(R\),L[2]) where \(S\) is computed via RandomObjectByList(C,L[1]).

2.5-8 RandomMorphismWithFixedRangeByInteger

Returns: a morphism in C

The arguments are an object S in an additive closure category C and an integer n. The output is RandomMorphismWithFixedSourceAndRangeByInteger(S,\(R\),\(1\)+Log2Int(n)) where \(S\) is computed via RandomObjectByInteger(C,n).

2.5-9 RandomMorphismByList

Returns: a morphism in C

The arguments are an additive closure category C and a list L consisiting of three lists. The output is RandomMorphismWithFixedSourceAndRangeByList(\(S\),\(R\),L[3])) where \(S\) and \(R\) are computed via RandomObjectByList(C,L[i]) for \(i=1,2\) respectively.

2.5-10 RandomMorphismByInteger

Returns: a morphism in C

The arguments are an additive closure category C and a non-negative integer n. The output is RandomMorphismWithFixedSourceAndRangeByInteger(\(S\),\(R\),\(1\)+Log2Int(n))) where \(S\) and \(R\) are computed via RandomObjectByInteger(C,n).

2.6 Global functions

2.6-1 NullMatImmutable

A (faster) version of NullMat returning an immutable matrix.

2.6-2 UnionOfRowsListList

Returns: a list of lists

Stacks the matrices (lists of lists) in the list L. The matrices must have nr_cols columns.

2.6-3 UnionOfColumnsListList

Returns: a list of lists

Augments the matrices (lists of lists) in the list L. The matrices must have nr_rows rows.

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