gap> LoadPackage( "ModulePresentationsForCAP", false );
true
gap> LoadPackage( "QuotientCategories", false );
true
gap> zz := HomalgRingOfIntegers( );;
gap> zz_pres := LeftPresentations( zz );
Category of left presentations of Z
gap> congruence_func := phi -> IsLiftable( phi, EpimorphismFromSomeProjectiveObject( Target( phi ) ) );;
gap> name := "QuotientCategory( Category of left presentations of Z ) by projectives";;
gap> quotient_cat := QuotientCategory(
>                   rec( underlying_category := zz_pres,
>                     name := name,
>                     congruence_func := congruence_func,
>                     nr_arguments_of_congruence_func := 1 ) );
QuotientCategory( Category of left presentations of Z ) by projectives
gap> Display( quotient_cat );
A CAP category with name QuotientCategory( Category of left presentations of Z ) by projectives:

23 primitive operations were used to derive 134 operations for this category
which algorithmically
* IsLinearCategoryOverCommutativeRing
* IsAdditiveCategory
gap> m := HomalgMatrix( [ [ -2, 0, 0, -1 ], [ 8, 0, 0, 4 ], [ -43, 2, 1, -17 ], [ 6, 0, 0, 3 ] ], 4, 4, zz );;
gap> M := AsLeftPresentation( zz_pres, m );
<An object in Category of left presentations of Z>
gap> IsProjective( M );
true
gap> quotient_M := M / quotient_cat;;
gap> IsZeroForObjects( quotient_M );
true