additive closure of Bézout domain is pre-abelian

AdditiveClosuresForCAP/PackageInfo.g

@@ -10,7 +10,7 @@ SetPackageInfo( rec(
 
 PackageName := "AdditiveClosuresForCAP",
 Subtitle := "Additive closures for pre-abelian categories",
-Version := "2025.06-01",
+Version := "2025.07-01",
 Date := (function ( ) if IsBound( GAPInfo.SystemEnvironment.GAP_PKG_RELEASE_DATE ) then return GAPInfo.SystemEnvironment.GAP_PKG_RELEASE_DATE; else return Concatenation( ~.Version{[ 1 .. 4 ]}, "-", ~.Version{[ 6, 7 ]}, "-01" ); fi; end)( ),
 License := "GPL-2.0-or-later",
 

AdditiveClosuresForCAP/examples/WeakHomomorphismTheorem.g

@@ -0,0 +1,52 @@
+#! @Chapter Example on additive closure
+
+#! @Section The category of rows over a Bézout domain is pre-abelian.
+
+#! @Example
+LoadPackage( "AdditiveClosuresForCAP", false );
+#! true
+zz := HomalgRingOfIntegers( );
+#! Z
+Zmat := CategoryOfRows( zz );
+#! Rows( Z )
+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
+mat := HomalgMatrix(
+               [ [  -36,   18,  -80,  -54,   28 ],
+                 [   90,  -45,  140,   45,  -40 ],
+                 [  -24,   12,  -32,   -4,    8 ],
+                 [   -6,    3,   -4,    5,    0 ] ], zz );;
+mor := mat / Zmat;
+#! <A morphism in Rows( Z )>
+split_epi := CoimageProjection( mor );
+#! <An epimorphism in Rows( Z )>
+IsSplitEpimorphism( split_epi );
+#! true
+split_mono := ImageEmbedding( mor );
+#! <A monomorphism in Rows( Z )>
+IsSplitMonomorphism( split_mono );
+#! true
+epimono := MorphismFromCoimageToImage( mor );
+#! <A morphism in Rows( Z )>
+IsEpimorphism( epimono );
+#! true
+IsMonomorphism( epimono );
+#! true
+LiftAlongMonomorphism( split_mono, AstrictionToCoimage( mor ) ) = epimono;
+#! true
+ColiftAlongEpimorphism( split_epi, CoastrictionToImage( mor ) ) = epimono;
+#! true
+PreCompose( [ split_epi, epimono, split_mono ] ) = mor;
+#! true
+#! @EndExample

AdditiveClosuresForCAP/gap/CategoryOfRows.gi

@@ -311,10 +311,14 @@ end );
 InstallGlobalFunction( INSTALL_FUNCTIONS_FOR_CATEGORY_OF_ROWS,
 
   function( category )
-    local ring, is_defined_over_field;
+    local ring, is_defined_over_bezout_domain, is_defined_over_field;
 
     ring := UnderlyingRing( category );
 
+    is_defined_over_bezout_domain :=
+      HasIsBezoutRing( ring ) and IsBezoutRing( ring ) and
+      HasIsIntegralDomain( ring ) and IsIntegralDomain( ring );
+
     is_defined_over_field := HasIsFieldForHomalg( ring ) and IsFieldForHomalg( ring );
 
     ## Well-defined for objects and morphisms
@@ -735,29 +739,9 @@ InstallGlobalFunction( INSTALL_FUNCTIONS_FOR_CATEGORY_OF_ROWS,
 
     end );
 
-    ## Abelian case
+    ## Pre-Abelian case
 
-    if is_defined_over_field then
-
-      ## enough projectives & injectives
-
-      ##
-      AddSomeProjectiveObject( category, { cat, obj } -> obj );
-
-      ##
-      AddEpimorphismFromSomeProjectiveObject( category, { cat, obj } -> IdentityMorphism( cat, obj ) );
-
-      ##
-      AddIsProjective( category, { cat, obj } -> true, 1 );
-
-      ##
-      AddSomeInjectiveObject( category, { cat, obj } -> obj );
-
-      ##
-      AddMonomorphismIntoSomeInjectiveObject( category, { cat, obj } -> IdentityMorphism( cat, obj ) );
-
-      ##
-      AddIsInjective( category, { cat, obj } -> true, 1 );
+    if is_defined_over_bezout_domain then
 
       ##
       AddKernelObject( category,
@@ -801,6 +785,12 @@ InstallGlobalFunction( INSTALL_FUNCTIONS_FOR_CATEGORY_OF_ROWS,
 
       end );
 
+    fi;
+
+    ## Abelian case
+
+    if is_defined_over_field then
+
       ##
       AddImageObject( category,
         function( cat, morphism )
@@ -839,6 +829,26 @@ InstallGlobalFunction( INSTALL_FUNCTIONS_FOR_CATEGORY_OF_ROWS,
 
       end );
 
+      ## enough projectives & injectives
+
+      ##
+      AddSomeProjectiveObject( category, { cat, obj } -> obj );
+
+      ##
+      AddEpimorphismFromSomeProjectiveObject( category, { cat, obj } -> IdentityMorphism( cat, obj ) );
+
+      ##
+      AddIsProjective( category, { cat, obj } -> true, 1 );
+
+      ##
+      AddSomeInjectiveObject( category, { cat, obj } -> obj );
+
+      ##
+      AddMonomorphismIntoSomeInjectiveObject( category, { cat, obj } -> IdentityMorphism( cat, obj ) );
+
+      ##
+      AddIsInjective( category, { cat, obj } -> true, 1 );
+
       ##
       AddSomeReductionBySplitEpiSummand( category,
         function( cat, alpha )

AdditiveClosuresForCAP/gap/CategoryOfRows_as_AdditiveClosure_RingAsCategory.gi

@@ -19,7 +19,7 @@ InstallMethod( CategoryOfRows_as_AdditiveClosure_RingAsCategory,
     [ "FinalizeCategory", true ],
   ],
   function( CAP_NAMED_ARGUMENTS, homalg_ring )
-    local ring_as_category, add, is_defined_over_field, object_constructor, modeling_tower_object_constructor, object_datum, modeling_tower_object_datum, morphism_constructor, modeling_tower_morphism_constructor, morphism_datum, modeling_tower_morphism_datum, wrapper;
+    local ring_as_category, add, is_defined_over_bezout_domain, is_defined_over_field, object_constructor, modeling_tower_object_constructor, object_datum, modeling_tower_object_datum, morphism_constructor, modeling_tower_morphism_constructor, morphism_datum, modeling_tower_morphism_datum, wrapper;
 
     ring_as_category := RING_AS_CATEGORY( homalg_ring );
 
@@ -37,6 +37,10 @@ InstallMethod( CategoryOfRows_as_AdditiveClosure_RingAsCategory,
 
     fi;
 
+    is_defined_over_bezout_domain :=
+      HasIsBezoutRing( homalg_ring ) and IsBezoutRing( homalg_ring ) and
+      HasIsIntegralDomain( homalg_ring ) and IsIntegralDomain( homalg_ring );
+
     is_defined_over_field := HasIsFieldForHomalg( homalg_ring ) and IsFieldForHomalg( homalg_ring );
 
     if is_defined_over_field then
@@ -47,6 +51,10 @@ InstallMethod( CategoryOfRows_as_AdditiveClosure_RingAsCategory,
 
         SetIsAbelianCategoryWithEnoughInjectives( add, true );
 
+    elif is_defined_over_bezout_domain then
+
+        SetIsPreAbelianCategory( add, true );
+
     fi;
 
     Finalize( add );

CAP/gap/DerivedMethods.autogen.gi

@@ -147,7 +147,7 @@ AddDerivationToCAP( CoimageProjection,
 
   function ( cat_1, alpha_1 )
     return PostCompose( cat_1, IsomorphismFromCokernelOfKernelToCoimage( cat_1, alpha_1 ), CokernelProjection( cat_1, KernelEmbedding( cat_1, alpha_1 ) ) );
-end : CategoryFilter := IsAbelianCategory,
+end : CategoryFilter := IsPreAbelianCategory,
       Weight := 1,
       is_autogenerated_by_CompilerForCAP := true );
 

CAP/gap/DerivedMethods.gi

@@ -500,7 +500,7 @@ AddDerivationToCAP( ImageEmbedding,
     return PreCompose( cat, IsomorphismFromImageObjectToKernelOfCokernel( cat, mor ),
                        image_embedding );
 
-  end : CategoryFilter := IsAbelianCategory ); ##FIXME: PreAbelian?
+  end : CategoryFilter := IsPreAbelianCategory );
 
 ##
 AddDerivationToCAP( CoastrictionToImage,
@@ -2935,7 +2935,7 @@ AddFinalDerivationBundle( "IsomorphismFromImageObjectToKernelOfCokernel as the i
     return IdentityMorphism( cat, kernel_of_cokernel );
 
   end,
-] : CategoryFilter := IsAbelianCategory );
+] : CategoryFilter := IsPreAbelianCategory );
 
 ##
 AddDerivationToCAP( MorphismFromCoimageToImageWithGivenObjects,
@@ -3012,7 +3012,7 @@ AddFinalDerivationBundle( "IsomorphismFromCoimageToCokernelOfKernel as the ident
     return IdentityMorphism( cat, cokernel_of_kernel );
 
   end,
-] : CategoryFilter := IsAbelianCategory );
+] : CategoryFilter := IsPreAbelianCategory );
 
 ## Final methods for initial object