VectorSpacesAllMethods.gi

if not IsBound( VectorSpacesConstructorsLoaded ) then
  
  ReadPackage( "CAP", "examples/testfiles/VectorSpacesConstructors.gi" );
  
fi;

if not IsCapCategory( vecspaces ) then

  vecspaces := CreateCapCategory( "VectorSpacesAllMethods" );
  
fi;

SetIsAbelianCategory( vecspaces, true );

AddIsCongruentForMorphisms( vecspaces,

  function( a, b )
  
    return a!.morphism = b!.morphism;
  
end );

AddIsEqualForObjects( vecspaces,

  function( a, b )
  
    return Dimension( a ) = Dimension( b );
  
end );

AddIsZeroForMorphisms( vecspaces,

  function( a )
    
    return IsZero( a!.morphism );
    
end );

AddAdditionForMorphisms( vecspaces,
                         
  function( a, b )
    
    return VectorSpaceMorphism( Source( a ), a!.morphism + b!.morphism, Range( a ) );
    
end );

AddAdditiveInverseForMorphisms( vecspaces,
                                
  function( a )
    
    return VectorSpaceMorphism( Source( a ), - a!.morphism, Range( a ) );
    
end );

AddZeroMorphism( vecspaces,
                     
  function( a, b )
    
    return VectorSpaceMorphism( a, HomalgZeroMatrix( Dimension( a ), Dimension( b ), VECTORSPACES_FIELD ), b );
    
end );

##
AddIdentityMorphism( vecspaces,
                     
  function( obj )

    return VectorSpaceMorphism( obj, HomalgIdentityMatrix( Dimension( obj ), VECTORSPACES_FIELD ), obj );
    
end );

##
AddPreCompose( vecspaces,

  function( mor_left, mor_right )
    local composition;

    composition := mor_left!.morphism * mor_right!.morphism;

    return VectorSpaceMorphism( Source( mor_left ), composition, Range( mor_right ) );

end );

##
AddZeroObject( vecspaces,

  function( )

    return QVectorSpace( 0 );

end );

##
AddLiftAlongMonomorphism( vecspaces,

  function( monomorphism, test_morphism )

    return VectorSpaceMorphism( Source( test_morphism ), RightDivide( test_morphism!.morphism, monomorphism!.morphism ), Source( monomorphism ) );

end );

##
AddColiftAlongEpimorphism( vecspaces,
  
  function( epimorphism, test_morphism )
    
    return VectorSpaceMorphism( Range( epimorphism ), LeftDivide( epimorphism!.morphism, test_morphism!.morphism ), Range( test_morphism ) );
    
end );

##
AddKernelObject( vecspaces,

  function( morphism )
    local homalg_matrix;

    homalg_matrix := morphism!.morphism;
  
    return QVectorSpace( NrRows( homalg_matrix ) - RowRankOfMatrix( homalg_matrix ) );

end );

##
AddKernelEmbedding( vecspaces,

  function( morphism )
    local kernel_emb, kernel_obj;
    
    kernel_emb := SyzygiesOfRows( morphism!.morphism );
    
    kernel_obj := QVectorSpace( NrRows( kernel_emb ) );
    
    return VectorSpaceMorphism( kernel_obj, kernel_emb, Source( morphism ) );
    
end );

##
AddKernelEmbeddingWithGivenKernelObject( vecspaces,

  function( morphism, kernel )
    local kernel_emb;

    kernel_emb := SyzygiesOfRows( morphism!.morphism );

    return VectorSpaceMorphism( kernel, kernel_emb, Source( morphism ) );

end );

##
AddCokernelObject( vecspaces,

  function( morphism )
    local homalg_matrix;

    homalg_matrix := morphism!.morphism;

    return QVectorSpace( NrColumns( homalg_matrix ) - RowRankOfMatrix( homalg_matrix ) );

end );


##
AddCokernelProjection( vecspaces,

  function( morphism )
    local cokernel_proj, cokernel_obj;

    cokernel_proj := SyzygiesOfColumns( morphism!.morphism );

    cokernel_obj := QVectorSpace( NrColumns( cokernel_proj ) );

    return VectorSpaceMorphism( Range( morphism ), cokernel_proj, cokernel_obj );

end );

##
AddCokernelProjectionWithGivenCokernelObject( vecspaces,

  function( morphism, cokernel )
    local cokernel_proj;

    cokernel_proj := SyzygiesOfColumns( morphism!.morphism );

    return VectorSpaceMorphism( Range( morphism ), cokernel_proj, cokernel );

end );

# ##
# AddCoproduct( vecspaces,
# 
#   function( object_product_list )
#     local dim;
#     
#     dim := Sum( List( object_product_list, c -> Dimension( c ) ) );
#     
#     return QVectorSpace( dim );
#   
# end );

##
## the user may assume that Length( object_product_list ) > 1
AddInjectionOfCofactorOfDirectSum( vecspaces,

  function( object_product_list, injection_number )
    local components, dim, dim_pre, dim_post, dim_cofactor, coproduct, number_of_objects, injection_of_cofactor;
    
    components := object_product_list;
    
    number_of_objects := Length( components );
    
    dim := Sum( components, c -> Dimension( c ) );
    
    dim_pre := Sum( components{ [ 1 .. injection_number - 1 ] }, c -> Dimension( c ) );
    
    dim_post := Sum( components{ [ injection_number + 1 .. number_of_objects ] }, c -> Dimension( c ) );
    
    dim_cofactor := Dimension( object_product_list[ injection_number ] );
    
    coproduct := QVectorSpace( dim );
    
    injection_of_cofactor := HomalgZeroMatrix( dim_cofactor, dim_pre ,VECTORSPACES_FIELD );
    
    injection_of_cofactor := UnionOfColumns( injection_of_cofactor, 
                                         HomalgIdentityMatrix( dim_cofactor, VECTORSPACES_FIELD ) );
    
    injection_of_cofactor := UnionOfColumns( injection_of_cofactor, 
                                         HomalgZeroMatrix( dim_cofactor, dim_post, VECTORSPACES_FIELD ) );
    
    return VectorSpaceMorphism( object_product_list[ injection_number ], injection_of_cofactor, coproduct );

end );

##
## the user may assume that Length( object_product_list ) > 1
AddInjectionOfCofactorOfDirectSumWithGivenDirectSum( vecspaces,

  function( object_product_list, injection_number, coproduct )
    local components, dim_pre, dim_post, dim_cofactor, number_of_objects, injection_of_cofactor;
    
    components := object_product_list;
    
    number_of_objects := Length( components );
    
    dim_pre := Sum( components{ [ 1 .. injection_number - 1 ] }, c -> Dimension( c ) );
    
    dim_post := Sum( components{ [ injection_number + 1 .. number_of_objects ] }, c -> Dimension( c ) );
    
    dim_cofactor := Dimension( object_product_list[ injection_number ] );
    
    injection_of_cofactor := HomalgZeroMatrix( dim_cofactor, dim_pre ,VECTORSPACES_FIELD );
    
    injection_of_cofactor := UnionOfColumns( injection_of_cofactor, 
                                         HomalgIdentityMatrix( dim_cofactor, VECTORSPACES_FIELD ) );
    
    injection_of_cofactor := UnionOfColumns( injection_of_cofactor, 
                                         HomalgZeroMatrix( dim_cofactor, dim_post, VECTORSPACES_FIELD ) );
    
    return VectorSpaceMorphism( object_product_list[ injection_number ], injection_of_cofactor, coproduct );

end );

##
AddUniversalMorphismFromDirectSum( vecspaces,

  function( diagram, sink )
    local dim, coproduct, components, universal_morphism, morphism;
    
    components := sink;
    
    dim := Sum( components, c -> Dimension( Source( c ) ) );
    
    coproduct := QVectorSpace( dim );
    
    universal_morphism := sink[1]!.morphism;
    
    for morphism in components{ [ 2 .. Length( components ) ] } do
    
      universal_morphism := UnionOfRows( universal_morphism, morphism!.morphism );
  
    od;
  
    return VectorSpaceMorphism( coproduct, universal_morphism, Range( sink[1] ) );
  
end );

##
AddUniversalMorphismFromDirectSumWithGivenDirectSum( vecspaces,

  function( diagram, sink, coproduct )
    local components, universal_morphism, morphism;
    
    components := sink;
    
    universal_morphism := sink[1]!.morphism;
    
    for morphism in components{ [ 2 .. Length( components ) ] } do
    
      universal_morphism := UnionOfRows( universal_morphism, morphism!.morphism );
  
    od;
  
    return VectorSpaceMorphism( coproduct, universal_morphism, Range( sink[1] ) );
  
end );

##
AddDirectSum( vecspaces,

  function( object_product_list )
    local dim;
    
    dim := Sum( List( object_product_list, c -> Dimension( c ) ) );
    
    return QVectorSpace( dim );
  
end );

#
# the user may assume that Length( object_product_list ) > 1
AddProjectionInFactorOfDirectSum( vecspaces,

  function( object_product_list, projection_number )
    local components, dim, dim_pre, dim_post, dim_factor, direct_product, number_of_objects, projection_in_factor;
    
    components := object_product_list;
    
    number_of_objects := Length( components );
    
    dim := Sum( components, c -> Dimension( c ) );
    
    dim_pre := Sum( components{ [ 1 .. projection_number - 1 ] }, c -> Dimension( c ) );
    
    dim_post := Sum( components{ [ projection_number + 1 .. number_of_objects ] }, c -> Dimension( c ) );
    
    dim_factor := Dimension( object_product_list[ projection_number ] );
    
    direct_product := QVectorSpace( dim );
    
    projection_in_factor := HomalgZeroMatrix( dim_pre, dim_factor, VECTORSPACES_FIELD );
    
    projection_in_factor := UnionOfRows( projection_in_factor, 
                                         HomalgIdentityMatrix( dim_factor, VECTORSPACES_FIELD ) );
    
    projection_in_factor := UnionOfRows( projection_in_factor, 
                                         HomalgZeroMatrix( dim_post, dim_factor, VECTORSPACES_FIELD ) );
    
    return VectorSpaceMorphism( direct_product, projection_in_factor, object_product_list[ projection_number ] );

end );

##
## the user may assume that Length( object_product_list ) > 1
AddProjectionInFactorOfDirectSumWithGivenDirectSum( vecspaces,

  function( object_product_list, projection_number, direct_product )
    local components, dim_pre, dim_post, dim_factor, number_of_objects, projection_in_factor;
    
    components := object_product_list;
    
    number_of_objects := Length( components );
    
    dim_pre := Sum( components{ [ 1 .. projection_number - 1 ] }, c -> Dimension( c ) );
    
    dim_post := Sum( components{ [ projection_number + 1 .. number_of_objects ] }, c -> Dimension( c ) );
    
    dim_factor := Dimension( object_product_list[ projection_number ] );
    
    projection_in_factor := HomalgZeroMatrix( dim_pre, dim_factor, VECTORSPACES_FIELD );
    
    projection_in_factor := UnionOfRows( projection_in_factor, 
                                         HomalgIdentityMatrix( dim_factor, VECTORSPACES_FIELD ) );
    
    projection_in_factor := UnionOfRows( projection_in_factor, 
                                         HomalgZeroMatrix( dim_post, dim_factor, VECTORSPACES_FIELD ) );
    
    return VectorSpaceMorphism( direct_product, projection_in_factor, object_product_list[ projection_number ] );

end );

AddUniversalMorphismIntoDirectSum( vecspaces,

  function( diagram, sink )
    local dim, direct_product, components, universal_morphism, morphism;
    
    components := sink;
    
    dim := Sum( components, c -> Dimension( Range( c ) ) );
    
    direct_product := QVectorSpace( dim );
    
    universal_morphism := sink[1]!.morphism;
    
    for morphism in components{ [ 2 .. Length( components ) ] } do
    
      universal_morphism := UnionOfColumns( universal_morphism, morphism!.morphism );
  
    od;
  
    return VectorSpaceMorphism( Source( sink[1] ), universal_morphism, direct_product );
  
end );

AddUniversalMorphismIntoDirectSumWithGivenDirectSum( vecspaces,

  function( diagram, sink, direct_product )
    local components, universal_morphism, morphism;
    
    components := sink;
    
    universal_morphism := sink[1]!.morphism;
    
    for morphism in components{ [ 2 .. Length( components ) ] } do
    
      universal_morphism := UnionOfColumns( universal_morphism, morphism!.morphism );
  
    od;
  
    return VectorSpaceMorphism( Source( sink[1] ), universal_morphism, direct_product );
  
end );

##
AddTerminalObject( vecspaces,

  function( )

    return QVectorSpace( 0 );

end );

##
AddUniversalMorphismIntoTerminalObject( vecspaces,

  function( sink )
    local morphism;

    morphism := VectorSpaceMorphism( sink, HomalgZeroMatrix( Dimension( sink ), 0, VECTORSPACES_FIELD ), QVectorSpace( 0 ) );

    return morphism;

end );

##
AddUniversalMorphismIntoTerminalObjectWithGivenTerminalObject( vecspaces,

  function( sink, terminal_object )
    local morphism;

    morphism := VectorSpaceMorphism( sink, HomalgZeroMatrix( Dimension( sink ), 0, VECTORSPACES_FIELD ), terminal_object );

    return morphism;

end );

##
AddInitialObject( vecspaces,

  function( )

    return QVectorSpace( 0 );

end );

##
AddUniversalMorphismFromInitialObject( vecspaces,

  function( source )
    local morphism;

    morphism := VectorSpaceMorphism( QVectorSpace( 0 ), HomalgZeroMatrix( 0, Dimension( source ), VECTORSPACES_FIELD ), source );

    return morphism;

end );

##
AddUniversalMorphismFromInitialObjectWithGivenInitialObject( vecspaces,

  function( source, initial_object )
    local morphism;

    morphism := VectorSpaceMorphism( initial_object, HomalgZeroMatrix( 0, Dimension( source ), VECTORSPACES_FIELD ), source );

    return morphism;

end );

##
AddIsWellDefinedForObjects( vecspaces,

  function( vectorspace )
  
    return IsHomalgRationalVectorSpaceRep( vectorspace ) and Dimension( vectorspace ) >= 0;
  
end );

##
AddIsWellDefinedForMorphisms( vecspaces,

  function( morphism )
    local matrix;
    
    if not IsHomalgRationalVectorSpaceMorphismRep( morphism ) then
      return false;
    fi;
    
    matrix := morphism!.morphism;
    
    return     IsHomalgMatrix( matrix )
           and NrRows( matrix ) = Dimension( Source( morphism ) )
           and NrColumns( matrix ) = Dimension( Range( morphism ) );
    
end );
# 
# AddIsZeroForObjects( vecspaces,
# 
#   function( obj )
#   
#     return Dimension( obj ) = 0;
#   
# end );
# 
# AddIsMonomorphism( vecspaces,
# 
#   function( morphism )
#   
#     return RowRankOfMatrix( morphism!.morphism ) = Dimension( Source( morphism ) );
#   
# end );
# 
# AddIsEpimorphism( vecspaces,
# 
#   function( morphism )
#   
#     return ColumnRankOfMatrix( morphism!.morphism ) = Dimension( Range( morphism ) );
#   
# end );
# 
# AddIsIsomorphism( vecspaces,
# 
#   function( morphism )
#   
#     return Dimension( Range( morphism ) ) = Dimension( Source( morphism ) ) 
#            and ColumnRankOfMatrix( morphism!.morphism ) = Dimension( Range( morphism ) );
#   
# end );

# ##
# AddImageObject( vecspaces,
# 
#   function( morphism )
#   
#     return QVectorSpace( RowRankOfMatrix( morphism!.morphism ) );
#   
# end );
# 

Finalize( vecspaces );