‣ 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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