gap> QQ := HomalgFieldOfRationalsInSingular();;
gap> R := QQ * "x,y,z";;
gap> rows := CategoryOfRows_as_AdditiveClosure_RingAsCategory( R );;
gap> M := HomalgMatrix( "[[x^2,4*y]]", 1, 2, R );;
gap> N := HomalgMatrix( "[[1,3*x], [2*y^3,5*y]]", 2, 2, R );;
gap> P := M * N;;
gap> o := CategoryOfRowsObject( rows, 1 );;
gap> A := o;;
gap> B := DirectSum( o, o );;
gap> alpha := CategoryOfRowsMorphism( rows, A, M, B );;
gap> IsWellDefined( alpha );
true
gap> beta := CategoryOfRowsMorphism( rows, B, N, B );;
gap> IsWellDefined( beta );
true
gap> gamma := PreCompose( alpha, beta );;
gap> IsWellDefined( gamma );
true
gap> UnderlyingMatrix( gamma ) = P;
true
gap> # compute Lift( gamma, beta ) via SolveLinearSystemInAbCategory for coverage of hom structure:
> delta := SolveLinearSystemInAbCategory( [ [ IdentityMorphism( A ) ] ], [ [ beta ] ], [ gamma ] )[1];;
gap> IsWellDefined( delta );
true
gap> IsCongruentForMorphisms( gamma, PreCompose( delta, beta ) );
true
gap> IsCongruentForMorphisms(
>     SumOfMorphisms( A, [ alpha, alpha, alpha ], B ),
>     AdditionForMorphisms( alpha, AdditionForMorphisms( alpha, alpha ) )
> );
true
gap> IsCongruentForMorphisms(
>     SumOfMorphisms( A, [ ], B ),
>     ZeroMorphism( A, B )
> );
true
gap> # E and EE are both occupied by GAP
> EEE := KoszulDualRing( R );;
gap> rows := CategoryOfRows_as_AdditiveClosure_RingAsCategory( EEE );;
gap> M := HomalgMatrix( "[[e0*e1,3*e0]]", 1, 2, EEE );;
gap> N := HomalgMatrix( "[[1,e0*e2], [2*e0*e1*e2,5*e2]]", 2, 2, EEE );;
gap> P := M * N;;
gap> o := CategoryOfRowsObject( rows, 1 );;
gap> A := o;;
gap> B := DirectSum( o, o );;
gap> alpha := CategoryOfRowsMorphism( rows, A, M, B );;
gap> IsWellDefined( alpha );
true
gap> beta := CategoryOfRowsMorphism( rows, B, N, B );;
gap> IsWellDefined( beta );
true
gap> gamma := PreCompose( alpha, beta );;
gap> IsWellDefined( gamma );
true
gap> UnderlyingMatrix( gamma ) = P;
true
gap> # compute Lift( gamma, beta ) via SolveLinearSystemInAbCategory for coverage of hom structure:
> delta := SolveLinearSystemInAbCategory( [ [ IdentityMorphism( A ) ] ], [ [ beta ] ], [ gamma ] )[1];;
gap> IsWellDefined( delta );
true
gap> IsCongruentForMorphisms( gamma, PreCompose( delta, beta ) );
true
gap> IsCongruentForMorphisms(
>     SumOfMorphisms( A, [ alpha, alpha, alpha ], B ),
>     AdditionForMorphisms( alpha, AdditionForMorphisms( alpha, alpha ) )
> );
true
gap> IsCongruentForMorphisms(
>     SumOfMorphisms( A, [ ], B ),
>     ZeroMorphism( A, B )
> );
true

gap> LoadPackage( "AdditiveClosuresForCAP", false );
true
gap> zz := HomalgRingOfIntegers( );
Z
gap> Zmat := CategoryOfRows( zz );
Rows( Z )
gap> Display( Zmat );
A CAP category with name Rows( Z ):

83 primitive operations were used to derive 425 operations for this category \
which algorithmically
* IsEquippedWithHomomorphismStructure
* IsLinearCategoryOverCommutativeRingWithFinitelyGeneratedFreeExternalHoms
* IsPreAbelianCategory
* IsRigidSymmetricClosedMonoidalCategory
* IsRigidSymmetricCoclosedMonoidalCategory
and furthermore mathematically
* IsSkeletalCategory
* IsStrictMonoidalCategory
gap> mat := HomalgMatrix(
>                [ [  -36,   18,  -80,  -54,   28 ],
>                  [   90,  -45,  140,   45,  -40 ],
>                  [  -24,   12,  -32,   -4,    8 ],
>                  [   -6,    3,   -4,    5,    0 ] ], zz );;
gap> mor := mat / Zmat;
<A morphism in Rows( Z )>
gap> split_epi := CoimageProjection( mor );
<An epimorphism in Rows( Z )>
gap> IsSplitEpimorphism( split_epi );
true
gap> split_mono := ImageEmbedding( mor );
<A monomorphism in Rows( Z )>
gap> IsSplitMonomorphism( split_mono );
true
gap> epimono := MorphismFromCoimageToImage( mor );
<A morphism in Rows( Z )>
gap> IsEpimorphism( epimono );
true
gap> IsMonomorphism( epimono );
true
gap> LiftAlongMonomorphism( split_mono, AstrictionToCoimage( mor ) ) = epimono;
true
gap> ColiftAlongEpimorphism( split_epi, CoastrictionToImage( mor ) ) = epimono;
true
gap> PreCompose( [ split_epi, epimono, split_mono ] ) = mor;
true