GAP (FreydCategoriesForCAP) - Chapter 12: Examples and Tests

gap> R := HomalgRingOfIntegers();;
gap> rows := CategoryOfRows( R );;
gap> adelman := AdelmanCategory( rows );;
gap> obj1 := CategoryOfRowsObject( 1, rows );;
gap> obj2 := CategoryOfRowsObject( 2, rows );;
gap> id := IdentityMorphism( obj2 );;
gap> alpha := CategoryOfRowsMorphism( obj1, HomalgMatrix( [ [ 1, 2 ] ], 1, 2, R ), obj2 );;
gap> beta := CategoryOfRowsMorphism( obj2, HomalgMatrix( [ [ 1, 2 ], [ 3, 4 ] ], 2, 2, R ), obj2 );;
gap> gamma := CategoryOfRowsMorphism( obj2, HomalgMatrix( [ [ 1, 3 ], [ 3, 4 ] ], 2, 2, R ), obj2 );;
gap> obj1_a := AsAdelmanCategoryObject( obj1 );;
gap> obj2_a := AsAdelmanCategoryObject( obj2 );;
gap> m := AsAdelmanCategoryMorphism( beta );;
gap> n := AsAdelmanCategoryMorphism( gamma );;
gap> IsWellDefined( m );
true
gap> # backwards compatibility
> IsIdenticalObj( MorphismDatum( m ), beta );
true
gap> IsCongruentForMorphisms( PreCompose( m, n ), PreCompose( n, m ) );
false
gap> IsCongruentForMorphisms( SubtractionForMorphisms( m, m ), ZeroMorphism( obj2_a, obj2_a ) );
true
gap> IsCongruentForMorphisms( ZeroObjectFunctorial( adelman ),
>                          PreCompose( UniversalMorphismFromZeroObject( obj1_a), UniversalMorphismIntoZeroObject( obj1_a ) ) 
>                         );
true
gap> d := [ obj1_a, obj2_a ];;
gap> pi1 := ProjectionInFactorOfDirectSum( d, 1 );;
gap> pi2 := ProjectionInFactorOfDirectSum( d, 2 );;
gap> id := IdentityMorphism( DirectSum( d ) );;
gap> iota1 := InjectionOfCofactorOfDirectSum( d, 1 );;
gap> iota2 := InjectionOfCofactorOfDirectSum( d, 2 );;
gap> IsCongruentForMorphisms( PreCompose( pi1, iota1 ) + PreCompose( pi2, iota2 ), id );
true
gap> IsCongruentForMorphisms( UniversalMorphismIntoDirectSum( d, [ pi1, pi2 ] ), id );
true
gap> IsCongruentForMorphisms( UniversalMorphismFromDirectSum( d, [ iota1, iota2 ] ), id );
true
gap> c := CokernelProjection( m );;
gap> c2 := CokernelProjection( c );;
gap> IsCongruentForMorphisms( c2, ZeroMorphism( Source( c2 ), Range( c2 ) ) );
true
gap> IsWellDefined( CokernelProjection( m ) );
true
gap> IsCongruentForMorphisms( CokernelColift( m, CokernelProjection( m ) ), IdentityMorphism( CokernelObject( m ) ) );
true
gap> k := KernelEmbedding( c );;
gap> IsZeroForMorphisms( PreCompose( k, c ) );
true
gap> IsCongruentForMorphisms( KernelLift( m, KernelEmbedding( m ) ), IdentityMorphism( KernelObject( m ) ) );
true

12.2 Adelman category basics for category of columns

gap> R := HomalgRingOfIntegers();;
gap> cols := CategoryOfColumns( R );;
gap> adelman := AdelmanCategory( cols );;
gap> obj1 := CategoryOfColumnsObject( 1, cols );;
gap> obj2 := CategoryOfColumnsObject( 2, cols );;
gap> id := IdentityMorphism( obj2 );;
gap> alpha := CategoryOfColumnsMorphism( obj1, HomalgMatrix( [ [ 1 ], [ 2 ] ], 2, 1, R ), obj2 );;
gap> beta := CategoryOfColumnsMorphism( obj2, HomalgMatrix( [ [ 1, 3 ], [ 2, 4 ] ], 2, 2, R ), obj2 );;
gap> gamma := CategoryOfColumnsMorphism( obj2, HomalgMatrix( [ [ 1, 3 ], [ 3, 4 ] ], 2, 2, R ), obj2 );;
gap> obj1_a := AsAdelmanCategoryObject( obj1 );;
gap> obj2_a := AsAdelmanCategoryObject( obj2 );;
gap> m := AsAdelmanCategoryMorphism( beta );;
gap> n := AsAdelmanCategoryMorphism( gamma );;
gap> IsWellDefined( m );
true
gap> IsCongruentForMorphisms( PreCompose( m, n ), PreCompose( n, m ) );
false
gap> IsCongruentForMorphisms( SubtractionForMorphisms( m, m ), ZeroMorphism( obj2_a, obj2_a ) );
true
gap> IsCongruentForMorphisms( ZeroObjectFunctorial( adelman ),
>                          PreCompose( UniversalMorphismFromZeroObject( obj1_a), UniversalMorphismIntoZeroObject( obj1_a ) ) 
>                         );
true
gap> d := [ obj1_a, obj2_a ];;
gap> pi1 := ProjectionInFactorOfDirectSum( d, 1 );;
gap> pi2 := ProjectionInFactorOfDirectSum( d, 2 );;
gap> id := IdentityMorphism( DirectSum( d ) );;
gap> iota1 := InjectionOfCofactorOfDirectSum( d, 1 );;
gap> iota2 := InjectionOfCofactorOfDirectSum( d, 2 );;
gap> IsCongruentForMorphisms( PreCompose( pi1, iota1 ) + PreCompose( pi2, iota2 ), id );
true
gap> IsCongruentForMorphisms( UniversalMorphismIntoDirectSum( d, [ pi1, pi2 ] ), id );
true
gap> IsCongruentForMorphisms( UniversalMorphismFromDirectSum( d, [ iota1, iota2 ] ), id );
true
gap> c := CokernelProjection( m );;
gap> c2 := CokernelProjection( c );;
gap> IsCongruentForMorphisms( c2, ZeroMorphism( Source( c2 ), Range( c2 ) ) );
true
gap> IsWellDefined( CokernelProjection( m ) );
true
gap> IsCongruentForMorphisms( CokernelColift( m, CokernelProjection( m ) ), IdentityMorphism( CokernelObject( m ) ) );
true
gap> k := KernelEmbedding( c );;
gap> IsZeroForMorphisms( PreCompose( k, c ) );
true
gap> IsCongruentForMorphisms( KernelLift( m, KernelEmbedding( m ) ), IdentityMorphism( KernelObject( m ) ) );
true

12.3 Basics based on category of rows

gap> LoadPackage( "FreydCategoriesForCAP", false );
true
gap> LoadPackage( "RingsForHomalg", false );
true
gap> R := HomalgRingOfIntegers();;
gap> cat := CategoryOfRows( R );;
gap> obj1 := CategoryOfRowsObject( 1, cat );;
gap> obj2 := CategoryOfRowsObject( 2, cat );;
gap> gamma := CategoryOfRowsMorphism( obj2, HomalgMatrix( [ [ 1, 1 ], [ 1, 1 ] ], 2, 2, R ), obj2 );;
gap> ## Freyd categories
> freyd := FreydCategory( cat );;
gap> IsAbelianCategory( freyd );
true
gap> obj_gamma := FreydCategoryObject( gamma );;
gap> u := RandomMorphismWithFixedSource( obj_gamma, [ [5], [5] ] );;
gap> IsWellDefined( u ) and IsEqualForObjects( Source(u), obj_gamma );
true
gap> u := RandomMorphismWithFixedSource( obj_gamma, 5 );;
gap> IsWellDefined( u ) and IsEqualForObjects( Source(u), obj_gamma );
true
gap> u := RandomMorphismWithFixedRange( obj_gamma, [ [5], [5] ] );;
gap> IsWellDefined( u ) and IsEqualForObjects( Range(u), obj_gamma );
true
gap> u := RandomMorphismWithFixedRange( obj_gamma, 5 );;
gap> IsWellDefined( u ) and IsEqualForObjects( Range(u), obj_gamma );
true
gap> obj_delta := RandomObject( freyd, [[10],[10],[10]] );;
gap> obj_delta := RandomObject( freyd, 10 );;
gap> u := RandomMorphismWithFixedSourceAndRange( obj_gamma, obj_delta, [ 5 ] );;
gap> IsWellDefined( u );
true
gap> IsEqualForObjects( Source(u), obj_gamma ) and IsEqualForObjects( Range(u), obj_delta );
true
gap> u := RandomMorphismWithFixedSourceAndRange( obj_gamma, obj_delta, 5 );;
gap> IsWellDefined( u );
true
gap> IsEqualForObjects( Source(u), obj_gamma ) and IsEqualForObjects( Range(u), obj_delta );
true
gap> IsWellDefined( RandomMorphism( freyd, 5 ) );
true
gap> IsWellDefined( RandomMorphism( freyd, [[[5],[5],[5]],[[5],[5],[5]],[1]] ) );
true
gap> f := FreydCategoryMorphism( obj_gamma, gamma, obj_gamma );;
gap> witness := MorphismWitness( f );;
gap> g := FreydCategoryMorphism( obj_gamma, ZeroMorphism( obj2, obj2 ), obj_gamma );;
gap> IsCongruentForMorphisms( f, g );;
gap> c := PreCompose( f, f );;
gap> s := g + g;;
gap> a := CategoryOfRowsMorphism( obj1, HomalgMatrix( [ [ 2 ] ], 1, 1, R ), obj1 );;
gap> Z2 := FreydCategoryObject( a );;
gap> id := IdentityMorphism( Z2 );;
gap> z := id + id + id;;
gap> d := DirectSumFunctorial( [ z, z, z ] );;
gap> pr2 := ProjectionInFactorOfDirectSum( [ Z2, Z2, Z2 ], 2 );;
gap> pr3 := ProjectionInFactorOfDirectSum( [ Z2, Z2, Z2 ], 3 );;
gap> UniversalMorphismIntoDirectSum( [ pr3, pr2 ] );;
gap> inj1 := InjectionOfCofactorOfDirectSum( [ Z2, Z2, Z2 ], 1 );;
gap> inj2 := InjectionOfCofactorOfDirectSum( [ Z2, Z2, Z2 ], 2 );;
gap> UniversalMorphismFromDirectSum( [ inj2, inj1 ] );;
gap> ZFree := obj1/freyd;;
gap> id := IdentityMorphism( ZFree );;
gap> z := id + id;;
gap> CokernelProjection( z );;
gap> CokernelColift( z, CokernelProjection( z ) );;
gap> UniversalMorphismFromZeroObjectWithGivenZeroObject( obj_gamma, ZeroObject( freyd ) );;
gap> MultiplyWithElementOfCommutativeRingForMorphisms( 2 / R, f );;
gap> S := HomalgFieldOfRationalsInSingular() * "x,y,z";;
gap> Rows_S := CategoryOfRows( S );;
gap> S3 := CategoryOfRowsObject( 3, Rows_S );;
gap> S1 := CategoryOfRowsObject( 1, Rows_S );;
gap> mor := CategoryOfRowsMorphism( S3, HomalgMatrix( "[x,y,z]", 3, 1, S ), S1 );;
gap> k := FreydCategoryObject( mor );;
gap> w := EpimorphismFromSomeProjectiveObjectForKernelObject( UniversalMorphismIntoZeroObject( k ) );;
gap> k := KernelEmbedding( w );;
gap> ColiftAlongEpimorphism( CokernelProjection( k ), CokernelProjection( k ) );;
gap> ## Homomorphism structures
> Z4 := FreydCategoryObject( AsCategoryOfRowsMorphism( HomalgMatrix( "[4]", 1, 1, R ), cat ) );;
gap> Z3 := FreydCategoryObject( AsCategoryOfRowsMorphism( HomalgMatrix( "[3]", 1, 1, R ), cat ) );;
gap> HomomorphismStructureOnObjects( Z4, Z2 );;
gap> HomomorphismStructureOnObjects( Z4, Z4 );;
gap> HomomorphismStructureOnObjects( Z2, Z4 );;
gap> HomomorphismStructureOnObjects( Z3, Z4 );;
gap> HomomorphismStructureOnMorphisms( IdentityMorphism( DirectSum( Z4, Z2, Z3 ) ), -IdentityMorphism( DirectSum( Z4, Z3 ) ) );;
gap> ## Lifts
> S2 := CategoryOfRowsObject( 2, Rows_S );;
gap> S4 := CategoryOfRowsObject( 4, Rows_S );;
gap> S1_freyd := AsFreydCategoryObject( S1 );;
gap> S2_freyd := AsFreydCategoryObject( S2 );;
gap> S3_freyd := AsFreydCategoryObject( S3 );;
gap> S4_freyd := AsFreydCategoryObject( S4 );;
gap> lift := FreydCategoryMorphism( S1_freyd, CategoryOfRowsMorphism( S1, HomalgMatrix( "[x]", 1, 1, S ), S1 ), S1_freyd );;
gap> gamma := FreydCategoryMorphism( S1_freyd, CategoryOfRowsMorphism( S1, HomalgMatrix( "[y]", 1,1, S ), S1 ), S1_freyd );;
gap> alpha := PreCompose( lift, gamma );;
gap> IsLiftable( alpha, gamma );
true
gap> Lift( alpha, gamma );;
gap> IsColiftable( lift, alpha );
true
gap> IsCongruentForMorphisms( PreCompose( lift, Colift( lift, alpha ) ), alpha );
true
gap> lift := FreydCategoryMorphism( S2_freyd, CategoryOfRowsMorphism( S2, HomalgMatrix( "[x,y,z,x^2,1,z+1]", 2, 3, S ), S3 ), S3_freyd );;
gap> gamma := FreydCategoryMorphism( S3_freyd, CategoryOfRowsMorphism( S3, HomalgMatrix( "[x,y,z,z+1, x^2,y^2,z^2,z^2+1, x^3,y^3,z^3,z^3+1]", 3,4, S ), S4 ), S4_freyd );;
gap> alpha := PreCompose( lift, gamma );;
gap> Lift( alpha, gamma );;
gap> Colift( lift, alpha );;
gap> IsCongruentForMorphisms( PreCompose( lift, Colift( lift, alpha ) ), alpha );;
gap> interpretation := InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure( gamma );;
gap> IsCongruentForMorphisms( gamma,
> InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism( Source( gamma ), Range( gamma ), interpretation ) );;
gap> ## Opposite
> HomomorphismStructureOnObjects( Opposite( Z4 ), Opposite( Z2 ) );;
gap> HomomorphismStructureOnObjects( Z2, Z4 );;
gap> interpretation := InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure( Opposite( gamma ) );;
gap> IsCongruentForMorphisms( Opposite( gamma ),
> InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism( Source( Opposite( gamma ) ), Range( Opposite( gamma ) ), interpretation ) );
true

12.4 Basics based on category of columns

gap> R := HomalgRingOfIntegers();;
gap> cat := CategoryOfColumns( R );;
gap> obj1 := CategoryOfColumnsObject( 1, cat );;
gap> obj2 := CategoryOfColumnsObject( 2, cat );;
gap> gamma := CategoryOfColumnsMorphism( obj2, HomalgMatrix( [ [ 1, 1 ], [ 1, 1 ] ], 2, 2, R ), obj2 );;
gap> ## Freyd categories
> freyd := FreydCategory( cat );;
gap> IsAbelianCategory( freyd );
true
gap> obj_gamma := FreydCategoryObject( gamma );;
gap> f := FreydCategoryMorphism( obj_gamma, gamma, obj_gamma );;
gap> witness := MorphismWitness( f );;
gap> g := FreydCategoryMorphism( obj_gamma, ZeroMorphism( obj2, obj2 ), obj_gamma );;
gap> IsCongruentForMorphisms( f, g );;
gap> c := PreCompose( f, f );;
gap> s := g + g;;
gap> a := CategoryOfColumnsMorphism( obj1, HomalgMatrix( [ [ 2 ] ], 1, 1, R ), obj1 );;
gap> Z2 := FreydCategoryObject( a );;
gap> id := IdentityMorphism( Z2 );;
gap> z := id + id + id;;
gap> d := DirectSumFunctorial( [ z, z, z ] );;
gap> pr2 := ProjectionInFactorOfDirectSum( [ Z2, Z2, Z2 ], 2 );;
gap> pr3 := ProjectionInFactorOfDirectSum( [ Z2, Z2, Z2 ], 3 );;
gap> UniversalMorphismIntoDirectSum( [ pr3, pr2 ] );;
gap> inj1 := InjectionOfCofactorOfDirectSum( [ Z2, Z2, Z2 ], 1 );;
gap> inj2 := InjectionOfCofactorOfDirectSum( [ Z2, Z2, Z2 ], 2 );;
gap> UniversalMorphismFromDirectSum( [ inj2, inj1 ] );;
gap> ZFree := AsFreydCategoryObject( obj1 );;
gap> id := IdentityMorphism( ZFree );;
gap> z := id + id;;
gap> CokernelProjection( z );;
gap> CokernelColift( z, CokernelProjection( z ) );;
gap> S := HomalgFieldOfRationalsInSingular() * "x,y,z";;
gap> Cols_S := CategoryOfColumns( S );;
gap> S3 := CategoryOfColumnsObject( 3, Cols_S );;
gap> S1 := CategoryOfColumnsObject( 1, Cols_S );;
gap> mor := CategoryOfColumnsMorphism( S3, HomalgMatrix( "[x,y,z]", 1, 3, S ), S1 );;
gap> k := FreydCategoryObject( mor );;
gap> w := EpimorphismFromSomeProjectiveObjectForKernelObject( UniversalMorphismIntoZeroObject( k ) );;
gap> k := KernelEmbedding( w );;
gap> ColiftAlongEpimorphism( CokernelProjection( k ), CokernelProjection( k ) );;
gap> ## Homomorphism structures
> Z4 := FreydCategoryObject( AsCategoryOfColumnsMorphism( HomalgMatrix( "[4]", 1, 1, R ), cat ) );;
gap> Z3 := FreydCategoryObject( AsCategoryOfColumnsMorphism( HomalgMatrix( "[3]", 1, 1, R ), cat ) );;
gap> HomomorphismStructureOnObjects( Z4, Z2 );;
gap> HomomorphismStructureOnObjects( Z4, Z4 );;
gap> HomomorphismStructureOnObjects( Z2, Z4 );;
gap> HomomorphismStructureOnObjects( Z3, Z4 );;
gap> HomomorphismStructureOnMorphisms( IdentityMorphism( DirectSum( Z4, Z2, Z3 ) ), -IdentityMorphism( DirectSum( Z4, Z3 ) ) );;
gap> ## Lifts
> S2 := CategoryOfColumnsObject( 2, Cols_S );;
gap> S4 := CategoryOfColumnsObject( 4, Cols_S );;
gap> S1_freyd := AsFreydCategoryObject( S1 );;
gap> S2_freyd := AsFreydCategoryObject( S2 );;
gap> S3_freyd := AsFreydCategoryObject( S3 );;
gap> S4_freyd := AsFreydCategoryObject( S4 );;
gap> lift := FreydCategoryMorphism( S1_freyd, CategoryOfColumnsMorphism( S1, HomalgMatrix( "[x]", 1, 1, S ), S1 ), S1_freyd );;
gap> gamma := FreydCategoryMorphism( S1_freyd, CategoryOfColumnsMorphism( S1, HomalgMatrix( "[y]", 1,1, S ), S1 ), S1_freyd );;
gap> alpha := PreCompose( lift, gamma );;
gap> Lift( alpha, gamma );;
gap> Colift( lift, alpha );;
gap> IsCongruentForMorphisms( PreCompose( lift, Colift( lift, alpha ) ), alpha );;
gap> lift := FreydCategoryMorphism( S2_freyd, CategoryOfColumnsMorphism( S2, HomalgMatrix( "[x,y,z,x^2,1,z+1]", 3, 2, S ), S3 ), S3_freyd );;
gap> gamma := FreydCategoryMorphism( S3_freyd, CategoryOfColumnsMorphism( S3, HomalgMatrix( "[x,y,z,z+1, x^2,y^2,z^2,z^2+1, x^3,y^3,z^3,z^3+1]", 4,3, S ), S4 ), S4_freyd );;
gap> alpha := PreCompose( lift, gamma );;
gap> Lift( alpha, gamma );;
gap> Colift( lift, alpha );;
gap> IsCongruentForMorphisms( PreCompose( lift, Colift( lift, alpha ) ), alpha );;
gap> interpretation := InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure( gamma );;
gap> IsCongruentForMorphisms( gamma,
> InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism( Source( gamma ), Range( gamma ), interpretation ) );;
gap> ## Opposite
> HomomorphismStructureOnObjects( Opposite( Z4 ), Opposite( Z2 ) );;
gap> HomomorphismStructureOnObjects( Z2, Z4 );;
gap> interpretation := InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure( Opposite( gamma ) );;
gap> IsCongruentForMorphisms( Opposite( gamma ),
> InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism( Source( Opposite( gamma ) ), Range( Opposite( gamma ) ), interpretation ) );
true

12.5 CoFreyd category

gap> LoadPackage( "FreydCategoriesForCAP", false );
true
gap> ZZZ := HomalgRingOfIntegers( );;
gap> rows := CategoryOfRows( ZZZ );;
gap> co_freyd := CoFreydCategory( rows );
CoFreyd( Rows( Z ) )
gap> rows_obj := CategoryOfRowsObject( rows, 1 );; # ZZZ^1
gap> co_freyd_obj := AsCoFreydCategoryObject( rows_obj );; # ZZZ^1 -> 0
gap> rows_mor := CategoryOfRowsMorphism( rows, rows_obj, HomalgMatrix( [ 2 ], 1, 1, ZZZ ), rows_obj );; # ZZZ^1 --2-> ZZZ^1
gap> co_freyd_mor := CoFreydCategoryMorphism( co_freyd_obj, rows_mor, co_freyd_obj );;
gap> IsWellDefined( co_freyd_mor );
true
gap> Display( KernelObject( co_freyd_mor ) );

--------------------------------
CoRelation morphism:
--------------------------------

Source: 
A row module over Z of rank 1

Matrix: 
[ [  2 ] ]

Range: 
A row module over Z of rank 1

A morphism in Rows( Z )


--------------------------------
General description:
--------------------------------

An object in CoFreyd( Rows( Z ) )

gap> Display( UnderlyingMorphism( KernelEmbedding( co_freyd_mor ) ) );
Source: 
A row module over Z of rank 1

Matrix: 
[ [  1 ] ]

Range: 
A row module over Z of rank 1

An identity morphism in Rows( Z )
gap> Display( CokernelObject( co_freyd_mor ) );

--------------------------------
CoRelation morphism:
--------------------------------

Source: 
A row module over Z of rank 0

Matrix: 
(an empty 0 x 0 matrix)

Range: 
A row module over Z of rank 0

A morphism in Rows( Z )


--------------------------------
General description:
--------------------------------

An object in CoFreyd( Rows( Z ) )

gap> Display( UnderlyingMorphism( CokernelProjection( co_freyd_mor ) ) );
Source: 
A row module over Z of rank 1

Matrix: 
(an empty 1 x 0 matrix)

Range: 
A row module over Z of rank 0

A morphism in Rows( Z )

12.6 Cokernel image closure in category of rows

gap> R := HomalgFieldOfRationalsInSingular() * "x,y,z";;
gap> RowsR := CategoryOfRows( R );;
gap> m := AsCategoryOfRowsMorphism( 
>     HomalgMatrix( "[[x],[y],[z]]", 3, 1, R ), RowsR
> );;
gap> mu := AsCokernelImageClosureMorphism( m );;
gap> C := CokernelObject( mu );;
gap> C2 := AsFinitelyPresentedCokernelImageClosureObject( m );;
gap> IsEqualForObjects( C, C2 );
true
gap> n := AsCategoryOfRowsMorphism( 
>     HomalgMatrix( "[[x,y],[y^2,z]]", 2, 2, R ), RowsR
> );;
gap> nu := AsCokernelImageClosureMorphism( n );;
gap> nu2 := PreCompose( nu, nu );;
gap> IsWellDefined( nu2 );
true
gap> IsCongruentForMorphisms( nu, nu2 );
false
gap> nu + nu;;
gap> nu2 - nu;;
gap> cat := CapCategory( nu );;
gap> ZeroObject( cat );;
gap> ZeroMorphism( Source( nu ), Source( mu ) );;
gap> UniversalMorphismIntoZeroObject( Source( nu ) );;
gap> UniversalMorphismFromZeroObject( Source( nu ) );;
gap> S := Source( mu );;
gap> DirectSum( [S, S, S ] );;
gap> DirectSumFunctorial( [ nu2, nu ] );;
gap> UniversalMorphismIntoDirectSum( [ nu, nu ] );;
gap> UniversalMorphismFromDirectSum( [ nu, nu ] );;
gap> ProjectionInFactorOfDirectSum( [ S, S, S ], 2 );;
gap> InjectionOfCofactorOfDirectSum( [ S, S, S, S ], 4 );;
gap> CokernelColift( nu, CokernelProjection( nu ) );;
gap> IsCongruentForMorphisms( nu, PreCompose( CoastrictionToImage( nu ), ImageEmbedding( nu ) ) );
true
gap> u := UniversalMorphismFromImage( nu, [ nu, IdentityMorphism( Range( nu ) ) ] );;
gap> IsWellDefined( u );
true
gap> IsCongruentForMorphisms( nu, PreCompose( CoastrictionToImage( nu ), u ) );
true
gap> IsCongruentForMorphisms( u, ImageEmbedding( nu ) );
true
gap> kernel := KernelObject( mu );;
gap> emb := KernelEmbedding( mu );;
gap> p := PreCompose( EpimorphismFromSomeProjectiveObject( kernel ), KernelEmbedding( mu ) );;
gap> KernelLift( mu, p );;
gap> LiftAlongMonomorphism( emb, p );;
gap> I_to_A := FunctorCokernelImageClosureToFreydCategory( RowsR );;
gap> A_to_I := FunctorFreydCategoryToCokernelImageClosure( RowsR );;
gap> ApplyFunctor( I_to_A, kernel );;
gap> ApplyFunctor( A_to_I, ApplyFunctor( I_to_A, kernel ) );;
gap> nu := NaturalIsomorphismFromIdentityToFinitePresentationOfCokernelImageClosureObject( RowsR );;
gap> mu := NaturalIsomorphismFromFinitePresentationOfCokernelImageClosureObjectToIdentity( RowsR );;
gap> IsCongruentForMorphisms(
>     IdentityMorphism( kernel ),
>     PreCompose( ApplyNaturalTransformation( nu, kernel ), ApplyNaturalTransformation( mu, kernel ) )
> );
true

12.7 Cokernel image closure in category of columns

gap> R := HomalgFieldOfRationalsInSingular() * "x,y,z";;
gap> ColsR := CategoryOfColumns( R );;
gap> m := AsCategoryOfColumnsMorphism( 
>      HomalgMatrix( "[[x],[y],[z]]", 1, 3, R ), ColsR
> );;
gap> mu := AsCokernelImageClosureMorphism( m );;
gap> C := CokernelObject( mu );;
gap> C2 := AsFinitelyPresentedCokernelImageClosureObject( m );;
gap> IsEqualForObjects( C, C2 );
true
gap> n := AsCategoryOfColumnsMorphism( 
>     HomalgMatrix( "[[x,y],[y^2,z]]", 2, 2, R ), ColsR
> );;
gap> nu := AsCokernelImageClosureMorphism( n );;
gap> nu2 := PreCompose( nu, nu );;
gap> IsWellDefined( nu2 );
true
gap> IsCongruentForMorphisms( nu, nu2 );
false
gap> nu + nu;;
gap> nu2 - nu;;
gap> cat := CapCategory( nu );;
gap> ZeroObject( cat );;
gap> ZeroMorphism( Source( nu ), Source( mu ) );;
gap> UniversalMorphismIntoZeroObject( Source( nu ) );;
gap> UniversalMorphismFromZeroObject( Source( nu ) );;
gap> S := Source( mu );;
gap> DirectSum( [S, S, S ] );;
gap> DirectSumFunctorial( [ nu2, nu ] );;
gap> UniversalMorphismIntoDirectSum( [ nu, nu ] );;
gap> UniversalMorphismFromDirectSum( [ nu, nu ] );;
gap> ProjectionInFactorOfDirectSum( [ S, S, S ], 2 );;
gap> InjectionOfCofactorOfDirectSum( [ S, S, S, S ], 4 );;
gap> CokernelColift( nu, CokernelProjection( nu ) );;
gap> IsCongruentForMorphisms( nu, PreCompose( CoastrictionToImage( nu ), ImageEmbedding( nu ) ) );
true
gap> u := UniversalMorphismFromImage( nu, [ nu, IdentityMorphism( Range( nu ) ) ] );;
gap> IsWellDefined( u );
true
gap> IsCongruentForMorphisms( nu, PreCompose( CoastrictionToImage( nu ), u ) );
true
gap> IsCongruentForMorphisms( u, ImageEmbedding( nu ) );
true
gap> kernel := KernelObject( mu );;
gap> emb := KernelEmbedding( mu );;
gap> p := PreCompose( EpimorphismFromSomeProjectiveObject( kernel ), KernelEmbedding( mu ) );;
gap> KernelLift( mu, p );;
gap> LiftAlongMonomorphism( emb, p );;
gap> I_to_A := FunctorCokernelImageClosureToFreydCategory( ColsR );;
gap> A_to_I := FunctorFreydCategoryToCokernelImageClosure( ColsR );;
gap> ApplyFunctor( I_to_A, kernel );;
gap> ApplyFunctor( A_to_I, ApplyFunctor( I_to_A, kernel ) );;
gap> nu := NaturalIsomorphismFromIdentityToFinitePresentationOfCokernelImageClosureObject( ColsR );;
gap> mu := NaturalIsomorphismFromFinitePresentationOfCokernelImageClosureObjectToIdentity( ColsR );;
gap> IsCongruentForMorphisms(
>     IdentityMorphism( kernel ),
>     PreCompose( ApplyNaturalTransformation( nu, kernel ), ApplyNaturalTransformation( mu, kernel ) )
> );
true

12.8 Adelman category basics

gap> R := HomalgRingOfIntegers();;
gap> RowsR := CategoryOfRows( R );;
gap> one := AsCategoryOfRowsMorphism( HomalgMatrix( [ [ 1 ] ], 1, 1, R ), RowsR );;
gap> two := AsCategoryOfRowsMorphism( HomalgMatrix( [ [ 2 ] ], 1, 1, R ), RowsR );;
gap> four := AsCategoryOfRowsMorphism( HomalgMatrix( [ [ 4 ] ], 1, 1, R ), RowsR );;
gap> source := AdelmanCategoryObject( two, two );;
gap> range := AdelmanCategoryObject( two, four );;
gap> mor := AdelmanCategoryMorphism( source, one, range );;
gap> IsZeroForMorphisms( mor );
false
gap> emb := EmbeddingFunctorIntoFreydCategory( RowsR );;
gap> ind := AdelmanCategoryFunctorInducedByUniversalProperty( emb );;
gap> IsZeroForMorphisms( ApplyFunctor( ind, mor ) );
true
gap> M := FreydCategoryObject( AsCategoryOfRowsMorphism( HomalgMatrix( [ [  2, 2, 2 ], [ 4, 4, 6 ] ], 2, 3, R ), RowsR ) );;
gap> as_tensor := EmbeddingFunctorOfFreydCategoryIntoAdelmanCategory( RowsR );;
gap> Mt := ApplyFunctor( as_tensor, M );;
gap> lsat := LeftSatelliteAsEndofunctorOfAdelmanCategory( RowsR );;
gap> rsat := RightSatelliteAsEndofunctorOfAdelmanCategory( RowsR );;
gap> torsion := ApplyFunctor( ind, ( ApplyFunctor( rsat, ApplyFunctor( lsat, Mt ) ) ) );;
gap> unit := UnitOfSatelliteAdjunctionOfAdelmanCategory( RowsR );;
gap> IsZeroForMorphisms( ApplyNaturalTransformation( unit, Mt ) );
true
gap> counit := CounitOfSatelliteAdjunctionOfAdelmanCategory( RowsR );;
gap> t := ApplyNaturalTransformation( counit, Mt );;

12.9 Grade filtration

The sequence of modules computed via satellites behaves in a way that is not understood in the case when the ring is not Auslander regular.

gap> R := HomalgFieldOfRationalsInSingular() * "x,y";;
gap> R := R/"x*y"/"x^2";;
gap> RowsR := CategoryOfRows( R );;
gap> Freyd := FreydCategory( RowsR );;
gap> mat := HomalgMatrix( "[x,y]", 1, 2, R );;
gap> M := mat/Freyd;;
gap> mu1 := GradeFiltrationNthMonomorphism( M, 1 );;
gap> IsMonomorphism( mu1 );
true
gap> IsZero( mu1 );
false
gap> IsEpimorphism( mu1 );
false
gap> mu2 := GradeFiltrationNthMonomorphism( M, 2 );;
gap> IsIsomorphism( mu2 );
true
gap> IsZero( mu2 );
false
gap> mu3 := GradeFiltrationNthMonomorphism( M, 3 );;
gap> IsIsomorphism( mu3 );
true
gap> IsZero( mu3 );
false
gap> mu4 := GradeFiltrationNthMonomorphism( M, 4 );;
gap> IsMonomorphism( mu4 );
false
gap> IsEpimorphism( mu4 );
true
gap> IsZero( mu4 );
false

gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;
gap> wmat := HomalgMatrix( "[ \
> x*y,  y*z,    z,        0,         0,    \
> x^3*z,x^2*z^2,0,        x*z^2,     -z^2, \
> x^4,  x^3*z,  0,        x^2*z,     -x*z, \
> 0,    0,      x*y,      -y^2,      x^2-1,\
> 0,    0,      x^2*z,    -x*y*z,    y*z,  \
> 0,    0,      x^2*y-x^2,-x*y^2+x*y,y^2-y \
> ]", 6, 5, Qxyz );;
gap> RowsR := CategoryOfRows( Qxyz );;
gap> Freyd := FreydCategory( RowsR );;
gap> Adel := AdelmanCategory( RowsR );;
gap> M := wmat/Freyd;;

We compute the grade sequence of functors (it turns out that on the level of functors, we don't get monos)

gap> M_tor := M/Adel;;
gap> Mu1 := GradeFiltrationNthNaturalTransformationComponent( M_tor, 1 );;
gap> IsZero( Mu1 );
false
gap> IsMonomorphism( Mu1 );
true
gap> Mu2 := GradeFiltrationNthNaturalTransformationComponent( M_tor, 2 );;
gap> IsZero( Mu2 );
false
gap> IsMonomorphism( Mu2 );
false
gap> Mu3 := GradeFiltrationNthNaturalTransformationComponent( M_tor, 3 );;
gap> IsZero( Mu3 );
false
gap> IsMonomorphism( Mu3 );
false
gap> Mu4 := GradeFiltrationNthNaturalTransformationComponent( M_tor, 4 );;
gap> IsZero( Mu4 );
true

We compute the grade sequence of modules (here, we really get monos and thus a filtration)

gap> mu1 := GradeFiltrationNthMonomorphism( M, 1 );;
gap> IsZero( mu1 );
false
gap> IsMonomorphism( mu1 );
true
gap> mu2 := GradeFiltrationNthMonomorphism( M, 2 );;
gap> IsZero( mu2 );
false
gap> IsMonomorphism( mu2 );
true
gap> mu3 := GradeFiltrationNthMonomorphism( M, 3 );;
gap> IsZero( mu3 );
false
gap> IsMonomorphism( mu3 );
true
gap> mu4 := GradeFiltrationNthMonomorphism( M, 4 );;
gap> IsZero( mu4 );
true

12.10 Homomorphisms between f.p. functors based on category of rows

gap> R := HomalgFieldOfRationalsInSingular() * "x,y,z";;
gap> Rows_R := CategoryOfRows( R );;
gap> R1 := CategoryOfRowsObject( 1, Rows_R );;
gap> R3 := CategoryOfRowsObject( 3, Rows_R );;
gap> alpha := CategoryOfRowsMorphism( R3, HomalgMatrix( "[x,y,z]", 3, 1, R ), R1 );;
gap> M := FreydCategoryObject( alpha );;
gap> c0 := CovariantExtAsFreydCategoryObject( M, 0 );;
gap> c1 := CovariantExtAsFreydCategoryObject( M, 1 );;
gap> c2 := CovariantExtAsFreydCategoryObject( M, 2 );;
gap> IsZeroForObjects( HomomorphismStructureOnObjects( c0, c2 ) ); # = Ext^2( M, M )
false

12.11 Homomorphisms between f.p. functors based on category of columns

gap> R := HomalgFieldOfRationalsInSingular() * "x,y,z";;
gap> Cols_R := CategoryOfColumns( R );;
gap> R1 := CategoryOfColumnsObject( 1, Cols_R );;
gap> R3 := CategoryOfColumnsObject( 3, Cols_R );;
gap> alpha := CategoryOfColumnsMorphism( R3, HomalgMatrix( "[x,y,z]", 1, 3, R ), R1 );;
gap> M := FreydCategoryObject( alpha );;
gap> c0 := CovariantExtAsFreydCategoryObject( M, 0 );;
gap> c1 := CovariantExtAsFreydCategoryObject( M, 1 );;
gap> c2 := CovariantExtAsFreydCategoryObject( M, 2 );;
gap> IsZeroForObjects( HomomorphismStructureOnObjects( c0, c2 ) ); # = Ext^2( M, M )
false

12.12 Matrices over ZP K

#@if IsPackageMarkedForLoading( "FinSetsForCAP", ">= 2023.07-03" )
gap> #Incidence matrix of our proset
> K := [ [1, 1, 1], [0, 1, 1], [0, 1, 1] ];;
gap> #Construction of a tower of categories
> CP_K := ProSetAsCategory( K );;
gap> ZZZ := HomalgRingOfIntegers( );;
gap> ZP_K := LinearClosure( ZZZ, CP_K, ReturnTrue );;
gap> RowsP_K := AdditiveClosure( ZP_K );;
gap> a := ProSetAsCategoryObject( 1, CP_K );;
gap> b := ProSetAsCategoryObject( 2, CP_K );;
gap> c := ProSetAsCategoryObject( 3, CP_K );;
gap> #Three random objects in the additive closure
> #Such that there exists morphisms from A->B and B->C:
> rand_coef := List( [ 1 .. 5 ], i -> Random( [ 2 .. 20 ] ) );;
gap> A1 := List( [ 1 .. rand_coef[ 1 ] ], i -> a );;
gap> A2 := List( [ 1 .. rand_coef[ 2 ] ], i -> b );;
gap> A := Concatenation( A1, A2 );;
gap> B1 := List( [ 1 .. rand_coef[ 3 ] ], i -> b );;
gap> B2 := List( [ 1 .. rand_coef[ 4 ] ], i -> c );;
gap> B :=  Concatenation( B1, B2 );;
gap> C := List([ 1 .. rand_coef[ 5 ] ], i -> c);;
gap> #A random lifting problem over ZP_K
> MA_B :=  List( [ 1 .. rand_coef[ 1 ] + rand_coef[ 2 ] ], i ->
>             List( [ 1 .. rand_coef[ 3 ] + rand_coef[ 4 ] ], j ->
>                 LinearClosureMorphism( LinearClosureObject( A[i], ZP_K ), [Random( [ -20 .. 20 ] )], [ProSetAsCategoryMorphism( A[i], B[j] )], LinearClosureObject( B[j], ZP_K ) )
>                 )
>               );;
gap> alpha := MA_B/RowsP_K;;
gap> MB_C :=  List( [ 1 .. rand_coef[ 3 ] + rand_coef[ 4 ] ], i ->
>             List( [ 1 .. rand_coef[ 5 ] ], j ->
>                 LinearClosureMorphism( LinearClosureObject( B[i], ZP_K ), [Random( [ -20 .. 20 ] )], [ProSetAsCategoryMorphism( B[i], C[j] )], LinearClosureObject( C[j], ZP_K ) )
>                 )
>              );;
gap> beta := MB_C/RowsP_K;;
gap> gamma := PreCompose( alpha, beta );;
gap> lift := Lift( gamma, beta );;
gap> PreCompose(lift, beta) = gamma;
true
#@fi

12.13 Testing if an object is projective, injective, bijective, and computing projective/injective dimensions

gap> R := HomalgRingOfIntegers();;
gap> rows := CategoryOfRows( R );;
gap> adelman := AdelmanCategory( rows );;
gap> rowobj := CategoryOfRowsObject( 1, rows );;
gap> alpha := ProjectionInFactorOfDirectSum( [ rowobj, rowobj ], 1 );;
gap> alpha := AsAdelmanCategoryMorphism( alpha );;
gap> obj := CokernelObject( alpha );;
gap> IsBijectiveObject( obj );
true
gap> beta := 2 * IdentityMorphism( rowobj );;
gap> beta := AsAdelmanCategoryMorphism( beta );;
gap> obj1 := CokernelObject( beta );;
gap> obj2 := KernelObject( beta );;
gap> obj3 := ImageObject( beta );;
gap> obj4 := HomologyObject( beta, beta );;
gap> IsBijectiveObject( obj1 );
false
gap> IsInjective( obj1 );
true
gap> ProjectiveDimension( obj1 );
2
gap> IsBijectiveObject( obj2 );
false
gap> IsProjective( obj2 );
true
gap> InjectiveDimension( obj2 );
2
gap> ProjectiveDimension( obj3 );
1
gap> InjectiveDimension( obj3 );
1
gap> ProjectiveDimension( obj4 );
2
gap> InjectiveDimension( obj4 );
2

12.14 Prosets

#@if IsPackageMarkedForLoading( "FinSetsForCAP", ">= 2023.07-03" )
gap> K := [ [1, 1, 1], [0, 1, 1], [0, 1, 1] ];;
gap> L := [ [1, 1, 0], [0, 1, 1], [0, 0, 1] ];;
gap> P_K := ProSetAsCategory(K);;
gap> #ProSetAsCategory(L);
gap> a := 1/P_K;;
gap> b := ProSetAsCategoryObject(2, P_K);;
gap> c := ProSetAsCategoryObject(3, P_K);;
gap> d := ProSetAsCategoryObject(4, P_K);;
gap> delta := ProSetAsCategoryMorphism(b, a);;
gap> IsWellDefined(a);
true
gap> IsWellDefined(d);
false
gap> IsWellDefined(delta);
false
gap> alpha := ProSetAsCategoryMorphism(a, b);;
gap> beta := ProSetAsCategoryMorphism(b, c);;
gap> gamma := ProSetAsCategoryMorphism(a, c);;
gap> gamma = PreCompose(alpha, beta);
true
gap> id_a := IdentityMorphism(a);;
gap> IsWellDefined(Inverse(alpha));
false
gap> beta*Inverse(beta) = IdentityMorphism(b);
true
gap> alpha = Lift(gamma, beta);
true
gap> IsLiftable(beta, gamma);
false
gap> Colift(alpha, gamma) = beta;
true
gap> alpha = HomStructure(a, b, HomStructure(alpha));
true
#@fi

12.15 Category of relations

gap> F := HomalgRingOfIntegers( 3 );;
gap> vec := CategoryOfRows( F );;
gap> rel := RelCategory( vec );;
gap> A := 1/vec/rel;;
gap> id := IdentityMorphism( A );;
gap> IsWellDefined( id );
true
gap> alpha := HomalgMatrix( "[ 1, 2 ]", 2, 1, F )/vec;;
gap> alpha_rel := alpha/rel;;
gap> alpha_rel_inv := rel/alpha;;
gap> beta := PreCompose( alpha_rel_inv, alpha_rel );;
gap> IsCongruentForMorphisms( beta, id );
true
gap> IsEqualForMorphisms( beta, id );
false
gap> R := HomalgFieldOfRationalsInSingular() * "t";;
gap> t := IndeterminatesOfPolynomialRing( R )[1];;
gap> cocycle := function( a, b, c ) local e; e := CoastrictionToImage( UniversalMorphismIntoDirectSum( [ ReversedArrow( c ), Arrow( c ) ] ) ); return t^RankOfObject( KernelObject( e ) ); end;;
#@if IsPackageMarkedForLoading( "FinSetsForCAP", ">= 2023.07-03" )
gap> T := TwistedLinearClosure( R, rel, cocycle );;
gap> gamma := beta/T;;
gap> delta := ZeroMorphism( 1/vec, 1/vec )/rel/T;;
gap> IsZeroForMorphisms( 3*gamma - 3*gamma );
true
gap> IsCongruentForMorphisms( delta, gamma );
false
gap> beta := PreCompose( alpha_rel_inv/T, alpha_rel/T );;
gap> IsZeroForMorphisms( beta - t * IdentityMorphism( Range( alpha_rel/T ) ) );
true
gap> IsZeroForMorphisms( ( gamma * delta ) * gamma - gamma * ( delta * gamma ) );
true
#@fi

12 Examples and Tests

12.1 Adelman category basics for category of rows