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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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LoadPackage( "CAP" );

LoadPackage( "MatricesForHomalg" );

#ProfileFunctionsInGlobalVariables( true );
#ProfileOperationsAndMethods( true );
#ProfileGlobalFunctions( true );

ProfileMethods( IsEqualForCache );

###################################
##
## Types and Representations
##
###################################

DeclareRepresentation( "IsHomalgRationalVectorSpaceRep",
                       IsCapCategoryObjectRep,
                       [ ] );

BindGlobal( "TheTypeOfHomalgRationalVectorSpaces",
        NewType( TheFamilyOfCapCategoryObjects,
                IsHomalgRationalVectorSpaceRep ) );

DeclareRepresentation( "IsHomalgRationalVectorSpaceMorphismRep",
                       IsCapCategoryMorphismRep,
                       [ ] );

BindGlobal( "TheTypeOfHomalgRationalVectorSpaceMorphism",
        NewType( TheFamilyOfCapCategoryMorphisms,
                IsHomalgRationalVectorSpaceMorphismRep ) );

###################################
##
## Attributes
##
###################################
                
DeclareAttribute( "Dimension",
                  IsHomalgRationalVectorSpaceRep );

#######################################
##
## Operations
##
#######################################

DeclareOperation( "QVectorSpace",
                  [ IsInt ] );

DeclareOperation( "VectorSpaceMorphism",
                  [ IsHomalgRationalVectorSpaceRep, IsObject, IsHomalgRationalVectorSpaceRep ] );

vecspaces := CreateCapCategory( "VectorSpaces" );

SetIsAbelianCategory( vecspaces, true );

VECTORSPACES_FIELD := HomalgFieldOfRationals( );

#######################################
##
## Categorical Implementations
##
#######################################

##
InstallMethod( QVectorSpace,
               [ IsInt ],
               
  function( dim )
    local space;
    
    space := rec( );
    
    ObjectifyWithAttributes( space, TheTypeOfHomalgRationalVectorSpaces,
                             Dimension, dim 
    );

    # is this the right place?
    Add( vecspaces, space );
    
    return space;
    
end );

##
InstallMethod( VectorSpaceMorphism,
                  [ IsHomalgRationalVectorSpaceRep, IsObject, IsHomalgRationalVectorSpaceRep ],
                  
  function( source, matrix, range )
    local morphism;

    if not IsHomalgMatrix( matrix ) then
    
      morphism := HomalgMatrix( matrix, Dimension( source ), Dimension( range ), VECTORSPACES_FIELD );

    else

      morphism := matrix;

    fi;

    morphism := rec( morphism := morphism );
    
    
    ObjectifyWithAttributes( morphism, TheTypeOfHomalgRationalVectorSpaceMorphism,
                             Source, source,
                             Range, range 
    );

    Add( vecspaces, morphism );
    
    return morphism;
    
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, [ ] ], 100 );