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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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