‣ IsAdditiveClosureCategory( object ) | ( filter ) |
Returns: true or false
The GAP category of additive closures of Ab-categories.
‣ IsAdditiveClosureObject( object ) | ( filter ) |
Returns: true or false
The GAP category of objects in additive closures of Ab-categories.
‣ IsAdditiveClosureMorphism( object ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in additive closures of Ab-categories.
‣ AdditiveClosure( C ) | ( attribute ) |
Returns: the category \(C^\oplus\)
The argument is an Ab-category \(C\). The output is its additive closure \(C^\oplus\).
‣ ADDITIVE_CLOSURE( C ) | ( operation ) |
Returns: the category \(C^\oplus\)
Same as AdditiveClosure (2.2-1), but as an operation instead of an attribute.
‣ AdditiveClosureObject( L, C^\oplus ) | ( operation ) |
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 \(C^\oplus\).
‣ AsAdditiveClosureObject( A ) | ( attribute ) |
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 \(\iota:C\to C^\oplus\).
‣ AdditiveClosureMorphism( A, M, B ) | ( operation ) |
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 \(M\).
‣ AsAdditiveClosureMorphism( alpha ) | ( attribute ) |
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 \(\iota:C\to C^\oplus\).
‣ InclusionFunctorInAdditiveClosure( C ) | ( attribute ) |
Returns: a functor \(C\to C^\oplus\)
The argument is an Ab-category \(C\). The output is the inclusion functor \(\iota:C\to C^\oplus\).
‣ ExtendFunctorToAdditiveClosures( F ) | ( attribute ) |
Returns: a functor \(C^\oplus \to D^\oplus\)
The argument is a functor \(F:C\to D\), and the output is the extension functor \(F^\oplus:C^\oplus \to D^\oplus\).
‣ ExtendFunctorWithAdditiveRangeToFunctorFromAdditiveClosureOfSource( F ) | ( attribute ) |
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 \(F^\oplus:C^\oplus \to D\).
‣ ExtendFunctorToAdditiveClosureOfSource( F ) | ( attribute ) |
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).
‣ ExtendNaturalTransformationToAdditiveClosureOfSource( eta ) | ( attribute ) |
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 \(\eta^\oplus:(F^\oplus:C^\oplus\to D)\to(G^\oplus:C^\oplus\to D)\).
‣ UnderlyingCategory( A ) | ( attribute ) |
Returns: the category \(C\)
The argument is some additive closure category \(A:=C^\oplus\). The output is \(C\).
‣ ObjectList( A ) | ( attribute ) |
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 \([A_1,\dots,A_m]\).
‣ MorphismMatrix( alpha ) | ( attribute ) |
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 \(\alpha\).
‣ NumberRows( alpha ) | ( attribute ) |
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.
‣ NumberColumns( alpha ) | ( attribute ) |
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.
‣ []( A, i ) | ( operation ) |
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(\(A\)).
‣ [( alpha, i, j ) | ( operation ) |
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 \/‣ \/( arg1, arg2 ) | ( operation ) |
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 \/‣ \/( arg1, arg2 ) | ( operation ) |
This is a convenience method for AsAdditiveClosureObject and AsAdditiveClosureMorphism.
‣ RandomObjectByList( C, L ) | ( operation ) |
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]).
‣ RandomObjectByInteger( C, n ) | ( operation ) |
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).
‣ RandomMorphismWithFixedSourceAndRangeByList( S, R, L ) | ( operation ) |
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).
‣ RandomMorphismWithFixedSourceAndRangeByInteger( S, R, n ) | ( operation ) |
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).
‣ RandomMorphismWithFixedSourceByList( S, L ) | ( operation ) |
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]).
‣ RandomMorphismWithFixedSourceByInteger( S, n ) | ( operation ) |
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).
‣ RandomMorphismWithFixedRangeByList( R, L ) | ( operation ) |
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]).
‣ RandomMorphismWithFixedRangeByInteger( R, n ) | ( operation ) |
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).
‣ RandomMorphismByList( C, L ) | ( operation ) |
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.
‣ RandomMorphismByInteger( C, n ) | ( operation ) |
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).
‣ NullMatImmutable( arg ) | ( function ) |
A (faster) version of NullMat returning an immutable matrix.
‣ UnionOfRowsListList( nr_cols, L ) | ( function ) |
Returns: a list of lists
Stacks the matrices (lists of lists) in the list L. The matrices must have nr_cols columns.
‣ UnionOfColumnsListList( nr_rows, L ) | ( function ) |
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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