d&i IsClosedPreSheafWRTCoproductCocones

FiniteCocompletions/gap/CategoryOfColimitQuivers.gd

@@ -118,6 +118,15 @@ CapJitAddTypeSignature( "FiniteColimitCompletionWithStrictCoproductsOfUnderlying
 
 end );
 
+#! @Description
+#!  The inputs are a colimit quiver <A>colim_quiver</A> in the category $C$
+#!  and a list <A>vertices</A> of objects in $C$.
+#!  The output is the full colimit subquiver on these objects.
+#! @Returns a monomorphism of colimit quivers
+#! @Arguments colim_quiver, vertices
+DeclareOperation( "EmbeddingOfFullColimitSubquiver",
+        [ IsObjectInCategoryOfColimitQuivers, IsList ] );
+
 ####################################
 #
 #! @Section Constructors

FiniteCocompletions/gap/CategoryOfColimitQuivers.gi

@@ -349,6 +349,65 @@ InstallMethod( CategoryOfPreSheavesOfUnderlyingCategory,
 
 end );
 
+##
+InstallMethod( CategoryOfPreSheavesOfUnderlyingCategory,
+        [ IsCategoryOfColimitQuivers ],
+
+  function( ColimitQuiversC )
+
+    return PreSheaves( UnderlyingCategory( ColimitQuiversC ) );
+
+end );
+
+##
+InstallOtherMethodForCompilerForCAP( EmbeddingOfFullColimitSubquiver,
+        [ IsCategoryOfColimitQuivers, IsObjectInCategoryOfColimitQuivers, IsList ],
+
+  function( ColimitQuiversC, colim_quiver, list_of_objects )
+    local C, vertices_arrows, vertices, arrows, positions, V, id, A, colim_subquiver, embedding;
+
+    C := UnderlyingCategory( ColimitQuiversC );
+
+    vertices_arrows := ObjectDatum( ColimitQuiversC, colim_quiver );
+
+    vertices := vertices_arrows[1];
+    arrows := vertices_arrows[2];
+
+    positions := List( vertices, vertex ->
+                       PositionProperty( list_of_objects, obj -> IsEqualForObjects( C, obj, vertex ) ) );
+
+    V := Filtered( [ 0 .. Length( vertices ) - 1 ], i -> IsInt( positions[1 + i] ) );
+
+    id := List( list_of_objects, obj -> IdentityMorphism( C, obj ) );
+
+    A := Filtered( [ 0 .. Length( arrows ) - 1 ], j ->
+                 ( arrows[1 + j][1] in V ) and ( arrows[1 + j][3] in V ) );
+
+    colim_subquiver := CreateColimitQuiver( ColimitQuiversC,
+                               Pair( vertices{1 + V}, arrows{1 + A} ) );
+
+    embedding := CreateMorphismOfColimitQuivers( ColimitQuiversC,
+                         colim_subquiver,
+                         Pair( Pair( V, id{positions{1 + V}} ),
+                               A ),
+                         colim_quiver );
+    #% CAP_JIT_DROP_NEXT_STATEMENT
+    SetIsMonomorphism( embedding, true );
+
+    return embedding;
+
+end );
+
+##
+InstallMethod( EmbeddingOfFullColimitSubquiver,
+        [ IsObjectInCategoryOfColimitQuivers, IsList ],
+
+  function( colim_quiver, list_of_objects )
+
+    return EmbeddingOfFullColimitSubquiver( CapCategory( colim_quiver ), colim_quiver, list_of_objects );
+
+end );
+
 ####################################
 #
 # View, Print, Display and LaTeX methods:

FpCategories/PackageInfo.g

@@ -10,7 +10,7 @@ SetPackageInfo( rec(
 
 PackageName := "FpCategories",
 Subtitle := "Finitely presented categories by generating quivers and relations",
-Version := "2024.03-03",
+Version := "2024.03-04",
 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",
 

FpCategories/gap/PathCategories.gd

@@ -173,5 +173,11 @@ DeclareAttribute( "ExternalHoms", IsPathCategory );
 #
 ####################################
 
+DeclareAttribute( "DecompositionIndicesOfMorphism",
+        IsPathCategoryMorphism );
+
+DeclareAttribute( "DecompositionOfMorphismInCategory",
+        IsPathCategoryMorphism );
+
 DeclareOperation( "ExtendFunctorToFpCategoryData",
         [ IsPathCategory, IsList, IsCapCategory ] );

FpCategories/gap/PathCategories.gi

@@ -1066,6 +1066,42 @@ InstallOtherMethod( ExternalHoms,
 
 end );
 
+##
+InstallOtherMethodForCompilerForCAP( DecompositionIndicesOfMorphism,
+        "for a path category and a morphism therein",
+        [ IsPathCategory, IsPathCategoryMorphism ],
+
+  function( C, mor )
+
+    return List( MorphismIndices( mor ), i -> -1 + i );
+
+end );
+
+##
+InstallMethod( DecompositionIndicesOfMorphism,
+        "for a morphism in a path category",
+        [ IsPathCategoryMorphism ],
+
+  function( mor )
+
+    return DecompositionIndicesOfMorphism( CapCategory( mor ), mor );
+
+end );
+
+##
+InstallMethod( DecompositionOfMorphismInCategory,
+        "for a morphism in a path category",
+        [ IsPathCategoryMorphism ],
+
+  function( mor )
+    local C;
+
+    C := CapCategory( mor );
+
+    return SetOfGeneratingMorphisms( C ){1 + DecompositionIndicesOfMorphism( mor )};
+
+end );
+
 ##
 InstallMethod( DatumOfCellAsEvaluatableString,
         [ IsPathCategoryMorphism, IsList ],

FpCategories/gap/Tools.gd

@@ -88,6 +88,24 @@ CapJitAddTypeSignature( "DefiningTripleOfUnderlyingQuiver", [ IsCapCategory ],
 
 end );
 
+#! @Description
+#!  Return the list of decompositions of the morphism <A>mor</A>
+#!  in the finitely presented, Hom-finite category $C$,
+#!  where each decomposition is a list in the generators in <C>SetOfGeneratorsOfCategory</C>( $C$ ).
+#! @Arguments mor
+#! @Returns a list of lists of morphisms
+DeclareAttribute( "AllDecompositionsOfMorphismInCategory",
+        IsCapCategoryMorphism );
+
+CapJitAddTypeSignature( "AllDecompositionsOfMorphismInCategory", [ IsCapCategory, IsCapCategoryMorphism ],
+  function ( input_types )
+
+    return CapJitDataTypeOfListOf(
+                   CapJitDataTypeOfListOf(
+                           CapJitDataTypeOfMorphismOfCategory( input_types[1].category ) ) );
+
+end );
+
 #! @Description
 #!  The data tables of the category <A>C</A>.
 #! @Arguments C

FpCategories/gap/Tools.gi

@@ -221,6 +221,90 @@ InstallMethodForCompilerForCAP( CoYonedaEmbeddingData,
 
 end );
 
+##
+InstallOtherMethodForCompilerForCAP( AllDecompositionsOfMorphismInCategory,
+        "for a f.p. category with decidable colifts and a morphism therein",
+        [ IsCapCategory and IsFinitelyPresentedCategory and IsCategoryWithDecidableColifts, IsCapCategoryMorphism ],
+
+  function( P, mor )
+    local source, gen_mors, pos, predicate, decompose_one_step, func, add_non_identity_colift, initial_value;
+
+    if IsEqualToIdentityMorphism( P, mor ) then
+        return [ ];
+    fi;
+
+    source := Source( mor );
+
+    gen_mors := SetOfGeneratingMorphisms( P );
+
+    pos := PositionProperty( gen_mors, gen -> IsEqualForMorphismsOnMor( P, gen, mor ) );
+
+    if IsInt( pos ) then
+        return [ [ gen_mors[pos] ] ];
+    fi;
+
+    predicate :=
+      function( decompositions, decompositions_new )
+
+        return Length( decompositions ) = Length( decompositions_new );
+
+    end;
+
+    decompose_one_step :=
+      function( decomposition )
+        local head, decompositions_of_last;
+
+        decompositions_of_last := AllDecompositionsOfMorphismInCategory( P, Last( decomposition ) );
+
+        if Length( decompositions_of_last ) = 0 then
+            return [ decomposition ];
+        else
+            head := decomposition{[ 1 .. Length( decomposition ) - 1 ]};
+
+            return List( decompositions_of_last, decomps -> Concatenation( head, decomps ) );
+        fi;
+
+    end;
+
+    func :=
+      function( decompositions )
+
+        return Concatenation( List( decompositions, decompose_one_step ) );
+
+    end;
+
+    add_non_identity_colift :=
+      function( gen, mor )
+        local colift;
+
+        colift := Colift( P, gen, mor );
+
+        if IsEqualToIdentityMorphism( P, colift ) then
+            return [ gen ];
+        else
+            return [ gen, colift ];
+        fi;
+
+    end;
+
+    initial_value :=
+      List( Filtered( gen_mors, gen -> IsEqualForObjects( P, Source( gen ), source ) and IsColiftable( P, gen, mor ) ), gen -> add_non_identity_colift( gen, mor ) );
+
+    return CapFixpoint( predicate, func, initial_value );
+
+end );
+
+##
+InstallMethod( AllDecompositionsOfMorphismInCategory,
+        "for a morphism in a f.p. category with decidable colifts",
+        [ IsCapCategoryMorphism ],
+
+  function( mor )
+
+    return AllDecompositionsOfMorphismInCategory( CapCategory( mor ), mor );
+
+end );
+
 if IsPackageMarkedForLoading( "Digraphs", ">= 1.3.1" ) then
 
 ##

FunctorCategories/PackageInfo.g

@@ -89,7 +89,7 @@ Dependencies := rec(
   NeededOtherPackages := [
                    [ "GAPDoc", ">= 1.5" ],
                    [ "ToolsForHomalg", ">= 2022.12-01" ],
-                   [ "CAP", ">= 2024.02-01" ],
+                   [ "CAP", ">= 2024.04-02" ],
                    [ "MonoidalCategories", ">= 2024.02-01" ],
                    [ "CartesianCategories", ">= 2024.02-05" ],
                    [ "ToolsForCategoricalTowers", ">= 2024.03-02" ],

FunctorCategories/examples/ColimitingCocone.g

@@ -0,0 +1,35 @@
+#! @Chunk ColimitingCocone
+
+#! @Example
+LoadPackage( "FunctorCategories", false );
+#! true
+q := "q(a,l,m,r,i,i1,i2,c,b)[ac:a->c,lc:l->c,mi:m->i,ic:i->c,ri1:r->i1,i1i2:i1->i2,i2c:i2->c,cb:c->b]";;
+q := RightQuiver( q );
+#! RightQuiver( "" );
+F := FreeCategory( q );
+#F := PathCategory( q );
+#! FreeCategory(
+#! RightQuiver( "" ) );
+Size( F );
+#! 7
+PSh := PreSheaves( F );
+#! PreSheaves( FreeCategory(
+#! RightQuiver( "" ) ) );
+#! SkeletalFinSets )
+Y := YonedaEmbeddingOfSourceCategory( PSh );
+#! Yoneda embedding functor
+coprd_cocone := [ F.c, [ F.lc, F.mi * F.ic, F.ri1 * F.i1i2 * F.i2c ] ];
+#coprd_cocone := [ F.c, [ F.ac, F.lc ] ];
+is_closed := G -> IsClosedPreSheafWRTCoproductCocones( PSh, G, [ coprd_cocone ] );
+#! function( G ) ... end
+G := Coproduct( [ PSh.a, PSh.a, PSh.l, PSh.m, PSh.r, PSh.r, PSh.c, PSh.b ] );
+#G := Coproduct( [ PSh.a, PSh.l, PSh.l ] );
+#! <An object in PreSheaves( FreeCategory(
+#!  RightQuiver( "" ) ),
+#!  SkeletalFinSets )>
+emb := EmbeddingIntoClosureOfPreSheafWRTCoproductCocones( G, [ coprd_cocone ] );
+Assert( 0, IsWellDefined( emb ) );
+Assert( 0, IsMonomorphism( emb ) );
+Assert( 0, Source( emb ) = G );
+Assert( 0, is_closed( Target( emb ) ) );
+#! @EndExample

FunctorCategories/examples/JoinCocone.g

@@ -0,0 +1,40 @@
+#! @Chunk JoinCocone
+
+#! @Example
+LoadPackage( "FunctorCategories", false );
+#! true
+q := FinQuiver( "q(a,l,r,i,c,b)[ac:a->c,lc:l->c,ri:r->i,ic:i->c,cb:c->b]" );
+#! FinQuiver( "q(a,l,r,i,c,b)[ac:a-≻c,lc:l-≻c,ri:r-≻i,ic:i-≻c,cb:c-≻b]" )
+F := PathCategory( q );
+#! PathCategory(
+#! FinQuiver( "q(a,l,r,i,c,b)[ac:a-≻c,lc:l-≻c,ri:r-≻i,ic:i-≻c,cb:c-≻b]" ) )
+Size( F );
+#! 16
+P := PosetOfCategory( F );
+#! PosetOfCategory( PathCategory(
+#! FinQuiver( "q(a,l,r,i,c,b)[ac:a-≻c,lc:l-≻c,ri:r-≻i,ic:i-≻c,cb:c-≻b]" ) ) )
+Size( P );
+#! 16
+digraphF := DigraphOfPoset( P );;
+digraphF!.vertexlabels := List( SetOfObjects( F ), String );
+#Splash( DotVertexLabelledDigraph( digraphF ) );
+#Splash( DotVertexLabelledDigraph( DigraphOfPoset( P ) ) );
+PSh := PreSheaves( P );
+#! PreSheaves( PosetOfCategory( PathCategory(
+#! FinQuiver( "q(a,l,r,i,c,b)[ac:a-≻c,lc:l-≻c,ri:r-≻i,ic:i-≻c,cb:c-≻b]" ) ) ),
+#! IntervalCategory )
+Y := YonedaEmbeddingOfSourceCategory( PSh );
+#! Yoneda embedding functor
+coproducts := [ [ P.c, [ P.l, P.r ] ] ];;
+#! [ [ An object in the poset given by: (c),
+#!     [ An object in the poset given by: (l),
+#!       An object in the poset given by: (r) ] ] ]
+PSh_J := ClosedPreSheavesWRTCoproducts( P, coproducts );;
+closed := SetOfObjects( PSh_J );;
+Assert( 0, Length( closed ) = 10 );
+psh_J := ModelingCategory( ModelingCategory( PSh_J ) );;
+Assert( 0, Length( SetOfObjects( psh_J ) ) = Length( SetOfObjects( PSh_J ) ) );
+is_closed := psh_J!.ObjectMembershipFunction;
+Assert( 0, ForAll( closed, obj -> is_closed( ObjectDatum( ModelingObject( PSh_J, obj ) ) ) ) );
+#Splash( DotVertexLabelledDigraph( PSh, [ Y, psh_J ], [ "grey", "blue", "red" ] : offset := 0 ) );
+#! @EndExample