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/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 ],

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
+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
+coprd_cocone := [ 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) ] ] ]
+is_closed := G -> IsClosedPreSheafWRTCoproducts( PSh, G, [ coprd_cocone ] );
+#! function( G ) ... end
+PShJ := FullSubcategoryByObjectMembershipFunction( PSh, is_closed );
+#Splash( DotVertexLabelledDigraph( PSh, [ Y, PShJ ], [ "grey", "blue", "red" ] ) );
+G := Coproduct( [ PSh.l, PSh.r ] );
+emb := EmbeddingIntoClosureOfPreSheafWRTCoproducts( G, [ coprd_cocone ] );
+#Assert( 0, IsWellDefined( emb ) );
+Assert( 0, IsMonomorphism( emb ) );
+Assert( 0, Source( emb ) = G );
+Assert( 0, is_closed( Target( emb ) ) );
+closed := List( SetOfObjects( PSh ), G -> Target( EmbeddingIntoClosureOfPreSheafWRTCoproducts( G, [ coprd_cocone ] ) ) );
+Assert( 0, Length( SetOfObjects( PShJ ) ) = Length( DuplicateFreeList( closed ) ) );
+#! @EndExample

FunctorCategories/gap/PreSheaves.gd

@@ -393,6 +393,10 @@ DeclareAttribute( "CoequalizerDataOfPreSheafUsingOptimizedCoYonedaLemma",
 DeclareAttribute( "AssociatedCoequalizerPairInPreSheaves",
         IsObjectInCategoryOfColimitQuivers );
 
+#! @Arguments F
+DeclareAttribute( "AssociatedCellInPreSheaves",
+        IsCellInCategoryOfColimitQuivers );
+
 #! @Arguments F
 DeclareAttribute( "AssociatedCoequalizerPairInPreSheaves",
         IsObjectInFiniteColimitCompletionWithStrictCoproducts );
@@ -417,3 +421,27 @@ DeclareOperation( "ApplyPreSheafToMorphismInFiniteStrictCoproductCompletion",
 #! @Arguments C
 DeclareAttribute( "CategoryOfInternalCategories",
         IsCapCategory );
+
+#! @Arguments PSh, presheaf, coproducts_cocones
+DeclareOperation( "IsClosedPreSheafWRTCoproductCocones",
+        [ IsObjectInPreSheafCategory, IsList ] );
+
+#! @Arguments PSh, presheaf, lists_of_cofactors
+DeclareOperation( "IsClosedPreSheafWRTCoproducts",
+        [ IsObjectInPreSheafCategory, IsList ] );
+
+#! @Arguments presheaf, lists_of_indices
+DeclareOperation( "IsClosedPreSheafWRTCoproductsByIndices",
+        [ IsObjectInPreSheafCategory, IsList ] );
+
+#! @Arguments presheaf, coproduct_cocone
+DeclareOperation( "EmbeddingIntoClosureOfPreSheafWRTCoproductCocone",
+        [ IsObjectInPreSheafCategory, IsList ] );
+
+#! @Arguments presheaf, coproducts_cocones
+DeclareOperation( "EmbeddingIntoClosureOfPreSheafWRTCoproductCocones",
+        [ IsObjectInPreSheafCategory, IsList ] );
+
+#! @Arguments presheaf, lists_of_cofactors
+DeclareOperation( "EmbeddingIntoClosureOfPreSheafWRTCoproducts",
+        [ IsObjectInPreSheafCategory, IsList ] );