d&i IsClosedPreSheafWRTCoproductCocones

FunctorCategories/gap/PreSheaves.gd

@@ -413,3 +413,15 @@ DeclareOperation( "ApplyPreSheafToMorphismInFiniteStrictCoproductCompletion",
 #! @Arguments C
 DeclareAttribute( "CategoryOfInternalCategories",
         IsCapCategory );
+
+DeclareOperation( "IsClosedPreSheafWRTCoproductCocones",
+        [ IsPreSheafCategory, IsObjectInPreSheafCategory, IsList ] );
+
+DeclareOperation( "IsClosedPreSheafWRTCoproducts",
+        [ IsPreSheafCategory, IsObjectInPreSheafCategory, IsList ] );
+
+DeclareOperation( "IsClosedPreSheafWRTCoproductsByIndices",
+        [ IsPreSheafCategory, IsObjectInPreSheafCategory, IsList ] );
+
+DeclareOperation( "CoproductCoconesClosureOperatorOfPreSheaf",
+        [ IsPreSheafCategory, IsObjectInPreSheafCategory, IsList ] );

FunctorCategories/gap/PreSheaves.gi

@@ -2775,7 +2775,10 @@ InstallMethodForCompilerForCAP( ApplyObjectInPreSheafCategoryOfFpEnrichedCategor
 
     else
 
-        Error( "the type of the source category `B` is not supported yet\n" );
+        morB_op := MorphismConstructor( B_op,
+                           SetOfObjects( B_op )[SafeUniquePositionProperty( SetOfObjects( B ), obj -> IsEqualForObjects( B, obj, Target( morB ) ) )],
+                           morB,
+                           SetOfObjects( B_op )[SafeUniquePositionProperty( SetOfObjects( B ), obj -> IsEqualForObjects( B, obj, Source( morB ) ) )] );
 
     fi;
 
@@ -4441,6 +4444,95 @@ InstallMethod( SimpleObjects,
 
 end );
 
+####################################
+#
+# Methods for operations
+#
+####################################
+
+##
+InstallMethod( IsClosedPreSheafWRTCoproductCocones,
+        [ IsPreSheafCategory, IsObjectInPreSheafCategory, IsList ],
+
+  function( PSh, F, list_of_coproduct_cones )
+    local H, coproducts, cocones;
+
+    H := Target( PSh );
+
+    coproducts := List( list_of_coproduct_cones, cocone -> cocone[1] );
+    cocones := List( list_of_coproduct_cones, cocone -> cocone[2] );
+
+    return ForAll( [ 1 .. Length( coproducts ) ], i ->
+                   IsIsomorphism( H,
+                           UniversalMorphismIntoDirectProduct( H,
+                                   List( cocones[i], morphism ->
+                                         ApplyObjectInPreSheafCategoryOfFpEnrichedCategoryToObject( PSh, F, Source( morphism ) ) ),
+                                   ApplyObjectInPreSheafCategoryOfFpEnrichedCategoryToObject( PSh, F, coproducts[i] ),
+                                   List( cocones[i], morphism ->
+                                         ApplyObjectInPreSheafCategoryOfFpEnrichedCategoryToMorphism( PSh, F, morphism ) ) ) ) );
+
+end );
+
+##
+InstallMethod( IsClosedPreSheafWRTCoproducts,
+        [ IsPreSheafCategory and IsThinCategory, IsObjectInPreSheafCategory, IsList ],
+
+  function( PSh, F, list_of_coproducts )
+    local P, coproducts, cocones, list_of_coproduct_cones;
+
+    P := Source( PSh );
+
+    if not CanCompute( P, "UniqueMorphism" ) then
+        Error( "the poset `P := Source( PSh )` cannot compute the categorical operation `UniqueMorphism`\n" );
+    fi;
+
+    coproducts := List( list_of_coproducts, cocone -> cocone[1] );
+    cocones := List( list_of_coproducts, cocone -> List( cocone[2], cofactor -> UniqueMorphism( P, cofactor, cocone[1] ) ) );
+
+    list_of_coproduct_cones := ListN( coproducts, cocones, { coproduct, cocone } -> [ coproduct, cocone ] );
+
+    return IsClosedPreSheafWRTCoproductCocones( PSh, F, list_of_coproduct_cones );
+
+end );
+
+##
+InstallMethod( IsClosedPreSheafWRTCoproductsByIndices,
+        [ IsPreSheafCategory and IsThinCategory, IsObjectInPreSheafCategory, IsList ],
+
+  function( PSh, F, list_of_coproducts_by_indices )
+    local offset, P, objects, coproducts, cofactors, list_of_coproducts;
+
+    offset := CAP_INTERNAL_RETURN_OPTION_OR_DEFAULT( "offset", 0 );
+
+    P := Source( PSh );
+
+    objects := SetOfObjects( P );
+
+    coproducts := List( list_of_coproducts_by_indices, cocone -> objects[offset + cocone[1]] );
+    cofactors := List( list_of_coproducts_by_indices, cocone -> List( cocone[2], i -> objects[offset + i] ) );
+
+    list_of_coproducts := ListN( coproducts, cofactors, { coproduct, cofactor } -> [ coproduct, cofactor ] );
+
+    return IsClosedPreSheafWRTCoproducts( PSh, F, list_of_coproducts );
+
+end );
+
+##
+InstallMethod( CoproductCoconesClosureOperatorOfPreSheaf,
+        [ IsPreSheafCategory, IsObjectInPreSheafCategory, IsList ],
+
+  function( PSh, F, list_of_coproduct_cones )
+    local coproducts, cocones, objects;
+
+    coproducts := List( list_of_coproduct_cones, cocone -> cocone[1] );
+    cocones := List( list_of_coproduct_cones, cocone -> cocone[2] );
+
+    objects := SetOfObjects( Source( PSh ) );
+
+    Error( );
+
+end );
+
 ####################################
 #
 # View, Print, Display and LaTeX methods:

Locales/examples/DifferenceCSL.g

@@ -0,0 +1,80 @@
+#! @Chunk DifferenceCSL
+
+# the assumption si <= mi leads to fewer meets and joins:
+
+#! @Example
+LoadPackage( "FunctorCategories", false );
+#! true
+q := "q(m0,s0,m1,s1,m2,s2,m0_x_s1,s0_u_m1,m1_x_s2,s1_u_m2)\
+[sm0:s0->m0,sm1:s1->m1,sm2:s2->m2,\
+p011:m0_x_s1->s1,\
+i101:m1->s0_u_m1,\
+p122:m1_x_s2->s2,\
+i212:m2->s1_u_m2,\
+mc0:m0_x_s1->s0,jc0:m0->s0_u_m1,\
+mc1:m1_x_s2->s1,jc1:m1->s1_u_m2]";;
+q := FinQuiver( q );
+#! FinQuiver( "q(m0,s0,m1,s1,m2,s2,m0_x_s1,s0_u_m1,m1_x_s2,s1_u_m2)
+#! [sm0:s0-≻m0,sm1:s1-≻m1,sm2:s2-≻m2,
+#!  p011:m0_x_s1-≻s1,
+#!  i101:m1-≻s0_u_m1,
+#!  p122:m1_x_s2-≻s2,
+#!  i212:m2-≻s1_u_m2,
+#!  mc0:m0_x_s1-≻s0,jc0:m0-≻s0_u_m1,
+#!  mc1:m1_x_s2-≻s1,jc1:m1-≻s1_u_m2]" )
+F := PathCategory( q );
+#! PathCategory( FinQuiver( "q(m0,s0,m1,s1,m2,s2,m0_x_s1,s0_u_m1,m1_x_s2,s1_u_m2)
+#! [sm0:s0-≻m0,sm1:s1-≻m1,sm2:s2-≻m2,
+#!  p011:m0_x_s1-≻s1,
+#!  i101:m1-≻s0_u_m1,
+#!  p122:m1_x_s2-≻s2,
+#!  i212:m2-≻s1_u_m2,
+#!  mc0:m0_x_s1-≻s0,jc0:m0-≻s0_u_m1,
+#!  mc1:m1_x_s2-≻s1,jc1:m1-≻s1_u_m2]" ) )
+Size( F );
+#! 35
+HomStructure( F.s0, F.s0_u_m1 );
+#! |1|
+HomStructure( F.s1, F.s1_u_m2 );
+#! |1|
+HomStructure( F.m0_x_s1, F.m0 );
+#! |1|
+HomStructure( F.m1_x_s2, F.m1 );
+#! |1|
+C := F /
+     [ [ F.p011 * F.sm1 * F.i101, F.mc0 * F.sm0 * F.jc0 ],
+       [ F.p122 * F.sm2 * F.i212, F.mc1 * F.sm1 * F.jc1 ] ];
+#! PathCategory( FinQuiver( "q(m0,s0,m1,s1,m2,s2,m0_x_s1,s0_u_m1,m1_x_s2,s1_u_m2)
+#! [sm0:s0-≻m0,sm1:s1-≻m1,sm2:s2-≻m2,
+#!  p011:m0_x_s1-≻s1,
+#!  i101:m1-≻s0_u_m1,
+#!  p122:m1_x_s2-≻s2,
+#!  i212:m2-≻s1_u_m2,
+#!  mc0:m0_x_s1-≻s0,jc0:m0-≻s0_u_m1,
+#!  mc1:m1_x_s2-≻s1,jc1:m1-≻s1_u_m2]" ) )
+#!  / [ p011⋅sm1⋅i101 = mc0⋅sm0⋅jc0, p122⋅sm2⋅i212 = mc1⋅sm1⋅jc1 ]
+Size( C );
+#! 33
+P := PosetOfCategory( F );;
+Size( P );
+#! 33
+digraphF := DigraphOfPoset( P );;
+digraphF!.vertexlabels := List( SetOfObjects( F ), String );
+#Splash( DotVertexLabelledDigraph( digraphF ) );
+PSh := PreSheaves( P );;
+digraphPSh := DigraphOfPoset( PSh );;
+#Splash( DotVertexLabelledDigraph( digraphPSh ) );
+presheaves := SetOfObjects( PSh );;
+Y := YonedaEmbeddingOfSourceCategory( PSh );
+digraphP := DigraphOfPoset( P );;
+digraphP!.vertexlabels := List( SetOfObjects( P ), o -> String( -1 + SafePosition( presheaves, Y( o ) ) ) );
+#Splash( DotVertexLabelledDigraph( digraphP ) );
+cocones := [ [ P.s0_u_m1, [ P.s0, P.m1 ] ],
+             [ P.s1_u_m2, [ P.s1, P.m2 ] ] ];
+is_closed := F -> IsClosedPreSheafWRTCoproducts( PSh, F, cocones );
+PShJ := FullSubcategoryByObjectMembershipFunction( PSh, is_closed );
+digraphPShJ := DigraphOfPoset( PShJ );;
+digraphPShJ!.vertexlabels := List( SetOfObjects( PShJ ), c -> String( -1 + SafePosition( presheaves, ObjectDatum( c ) ) ) );
+#Splash( DotVertexLabelledDigraph( digraphPShJ ) );
+
+#! @EndExample

Locales/examples/FreeBooleanAlgebra.g

@@ -0,0 +1,71 @@
+#! @Chunk FreeBooleanAlgebra
+
+#! Construct the free boolean algebra on two generators with its $2^(2^2)$ elements
+#! in two steps.
+
+#! @Example
+LoadPackage( "FunctorCategories", false );
+#! true
+pq := FinQuiver( "quiver(p,q)" );
+#! FinQuiver( "quiver(p,q)[]" )
+F := PathCategory( pq );
+#! PathCategory( FinQuiver( "quiver(p,q)[]" ) )
+P := PosetOfCategory( F );
+#! PosetOfCategory( PathCategory( FinQuiver( "quiver(p,q)[]" ) ) )
+Length( SetOfObjects( P ) );
+#! 2
+Dist := FreeDistributiveCompletion( P );
+#! FreeDistributiveCompletion(
+#! PosetOfCategory( PathCategory( FinQuiver( "quiver(p,q)[]" ) ) ) )
+Display( Dist );
+#! A CAP category with name FreeDistributiveCompletion(
+#! PosetOfCategory( PathCategory( FinQuiver( "quiver(p,q)[]" ) ) ) ):
+#! 
+#! 54 primitive operations were used to derive 250 operations for this category
+#! which algorithmically
+#! * IsFiniteCategory
+#! * IsEquippedWithHomomorphismStructure
+#! * IsDistributiveLattice
+#! and not yet algorithmically
+#! * IsBiHeytingAlgebra
+Length( SetOfObjects( Dist ) );
+#! 6
+digraphDist := DigraphOfPoset( Dist );;
+#Splash( DotVertexLabelledDigraph( digraphDist ) );
+DiffbCSL := MeetSemilatticeOfSingleDifferences( Dist );
+#! MeetSemilatticeOfSingleDifferences( FreeDistributiveCompletion(
+#! PosetOfCategory( PathCategory( FinQuiver( "quiver(p,q)[]" ) ) ) ) )
+Length( SetOfObjects( DiffbCSL ) );
+#! 13
+digraphDiffbCSL := DigraphOfPoset( DiffbCSL );;
+#Splash( DotVertexLabelledDigraph( digraphDiffbCSL ) );
+PSh := PreSheaves( DiffbCSL );
+#! PreSheaves( MeetSemilatticeOfSingleDifferences( FreeDistributiveCompletion(
+#! PosetOfCategory( PathCategory( FinQuiver( "quiver(p,q)[]" ) ) ) ) ),
+#! IntervalCategory )
+presheaves := SetOfObjects( PSh );;
+Length( presheaves );
+#! 85
+#digraphPSh := DigraphOfPoset( PSh );;
+#Splash( DotVertexLabelledDigraph( digraphPSh ) );
+cocones := [ [ 0, [ ] ],
+             [ 3, [ 5, 7 ] ],
+             [ 2, [ 5, 9 ] ],
+             [ 6, [ 7, 10 ] ],
+             [ 8, [ 9, 10 ] ],
+             [ 12, [ 7, 9 ] ],
+             [ 4, [ 3, 2, 12 ] ],
+             [ 11, [ 6, 8, 12 ] ],
+             [ 1, [ 4, 11 ] ] ];;
+is_closed := F -> IsClosedPreSheafWRTCoproductsByIndices( PSh, F, cocones : offset := 1 );
+PShJ := FullSubcategoryByObjectMembershipFunction( PSh, is_closed );
+#! FullSubcategoryByObjectMembershipFunction( PreSheaves(
+#! MeetSemilatticeOfSingleDifferences( FreeDistributiveCompletion(
+#! PosetOfCategory( PathCategory( FinQuiver( "quiver(p,q)[]" ) ) ) ) ),
+#! IntervalCategory ), ObjectMembershipFunction )
+digraphPShJ := DigraphOfPoset( PShJ );;
+digraphPShJ!.vertexlabels := List( SetOfObjects( PShJ ), c -> String( -1 + SafePosition( presheaves, ObjectDatum( c ) ) ) );
+Length( SetOfObjects( PShJ ) );
+#! 16
+#Splash( DotVertexLabelledDigraph( digraphPShJ ) );
+#! @EndExample

Locales/gap/Proset.gi

@@ -75,6 +75,36 @@ InstallOtherMethod( Size,
 
 end );
 
+##
+InstallOtherMethod( CapFunctor,
+        "for a thin category, two lists, and the interval category",
+        [ IsThinCategory and IsFiniteCategory, IsList, IsList, IsIntervalCategory ],
+
+  function( P, imgs_of_objs, imgs_of_gmors, interval_category )
+    local F;
+
+    F := CapFunctor( Concatenation( "Functor from ", Name( P ), " -> ", Name( interval_category ) ), P, interval_category );
+
+    AddObjectFunction( F,
+      function ( obj )
+
+        return imgs_of_objs[SafeUniquePositionProperty( SetOfObjects( P ), o -> IsEqualForObjects( P, o, obj ) )];
+
+    end );
+
+    AddMorphismFunction( F,
+      function ( F_s, mor, F_t )
+
+        return UniqueMorphism( interval_category,
+                       F_s,
+                       F_t );
+
+    end );
+
+    return F;
+
+end );
+
 ##
 InstallMethod( Display,
         "for an object in a proset",