AddProjectionOntoCoequalizerOfIdentityAndAutomorphisms

Author: mohamed-barakat

PackageInfo.g

@@ -10,7 +10,7 @@ SetPackageInfo( rec(
 
 PackageName := "FinGSetsForCAP",
 Subtitle := "The (skeletal) elementary topos of finite G-sets",
-Version := "2026.05-02",
+Version := "2026.05-03",
 
 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",

examples/SkeletalCategoryOfTransitiveLeftGSets.g

@@ -14,14 +14,14 @@ Display( TS3 );
 #! A CAP category with name SkeletalCategoryOfTransitiveLeftGSets( S3 ) \
 #! with 4 objects:
 #! 
-#! 24 primitive operations were used to derive 80 operations for this category \
+#! 25 primitive operations were used to derive 80 operations for this category \
 #! which algorithmically
 #! * IsCategoryWithDecidableColifts
 #! * IsCategoryWithDecidableLifts
 #! * IsCategoryWithTerminalObject
 #! * IsFiniteCategory
-#! * IsCategoryWithCoequalizers
 #! * IsEquippedWithHomomorphismStructure
+#! * IsCategoryWithCoequalizers
 #! and not yet algorithmically
 #! * IsFinitelyPresentedCategory
 #! and furthermore mathematically

examples/notebooks/TS4.ipynb

@@ -112,7 +112,7 @@
      "text": [
       "A CAP category with name SkeletalCategoryOfTransitiveLeftGSets( S4 ) with 11 objects:\n",
       "\n",
-      "24 primitive operations were used to derive 80 operations for this category which algorithmically\n",
+      "25 primitive operations were used to derive 80 operations for this category which algorithmically\n",
       "* IsCategoryWithDecidableColifts\n",
       "* IsCategoryWithDecidableLifts\n",
       "* IsCategoryWithTerminalObject\n",

gap/SkeletalCategoryOfTransitiveLeftGSets.gi

@@ -342,6 +342,46 @@ InstallMethod( SkeletalCategoryOfTransitiveLeftGSets,
 
     end );
 
+    ##
+    AddProjectionOntoCoequalizerOfIdentityAndAutomorphisms( SkeletalTransitiveGSets,
+      function ( SkeletalTransitiveGSets, target, diagram )
+        local G, U, objects, t, l, gs, Ucoeq, index, cards, positions, pos, g, coeq;
+        
+        G := UnderlyingGroup( SkeletalTransitiveGSets );
+        
+        U := RepresentativesOfSubgroupsUpToConjugation( SkeletalTransitiveGSets );
+        
+        objects := SetOfObjects( SkeletalTransitiveGSets );
+        
+        t := ObjectNumber( target );
+        
+        l := Length( diagram );
+        
+        gs := List( [ 1 .. l ], i -> UnderlyingGroupElement( diagram[i] ) );
+        
+        Ucoeq := Subgroup( G, Concatenation( GeneratorsOfGroup( U[t] ), gs ) );
+        
+        index := Index( G, Ucoeq );
+        
+        cards := CardinalitiesOfObjects( SkeletalTransitiveGSets );
+        
+        positions := Filtered( [ 1 .. NumberOfObjects( SkeletalTransitiveGSets ) ], i -> cards[i] = index );
+        
+        ## pos := SafeUniqueEntry( positions, c -> IsConjugate( G, Ucoeq, U[c] ) );
+        ## but for performance we use:
+        pos := SafeFirst( positions, c -> IsConjugate( G, Ucoeq, U[c] ) );
+        
+        g := RepresentativeAction( G, Ucoeq, U[pos] );
+        
+        coeq := objects[pos];
+        
+        return MorphismConstructor( SkeletalTransitiveGSets,
+                       target,
+                       g,
+                       coeq );
+        
+    end );
+    
     ##
     AddIsHomSetInhabited( SkeletalTransitiveGSets,
       function ( SkeletalTransitiveGSets, source, target )