‣ UnderlyingCategory( ColimitQuivers ) | ( attribute ) |
Return the category \(C\) underlying the category of colimit quivers ColimitQuivers := CategoryOfColimitQuivers( \(C\) ).
‣ EmbeddingOfUnderlyingCategory( ColimitQuivers ) | ( attribute ) |
Returns: a CAP functor
The full embedding functor from the category \(C\) underlying the category of colimit quivers ColimitQuivers into ColimitQuivers.
‣ DefiningPairOfColimitQuiver( colimit_quiver ) | ( attribute ) |
‣ DefiningPairOfColimitQuiverMorphism( colimit_quiver_morphism ) | ( attribute ) |
‣ FiniteColimitCompletionWithStrictCoproductsOfUnderlyingCategory( PSh ) | ( attribute ) |
Returns: a CAP category
Given the presheaf category PSh=PreSheaves( \(C\), \(V\) ), return the ambient category CoequalizerCompletion( AssociatedFiniteStrictCoproductCompletionOfSourceCategory( PSh ) ).
‣ CategoryOfColimitQuivers( C ) | ( attribute ) |
Returns: a CAP category
Construct the category colimit quivers in the category C.
gap> LoadPackage( "FunctorCategories", ">= 2024.03-18", false ); true gap> FinBouquets; FinBouquets gap> Chat := ModelingCategory( FinBouquets ); FiniteCocompletion( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ) gap> ColimitQuiversC := CategoryOfColimitQuivers( > UnderlyingCategory( FinBouquets ) ); CategoryOfColimitQuivers( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ) gap> P := ColimitQuiversC.P; <A projective object in CategoryOfColimitQuivers( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) )> gap> Display( P ); [ [ <(P)> ], [ ] ] An object in CategoryOfColimitQuivers( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ) given by the above data gap> L := ColimitQuiversC.L; <A projective object in CategoryOfColimitQuivers( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) )> gap> Display( L ); [ [ <(L)> ], [ ] ] An object in CategoryOfColimitQuivers( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ) given by the above data gap> b := ColimitQuiversC.b; <A morphism in CategoryOfColimitQuivers( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) )> gap> Display( b ); Source: [ [ <(P)> ], [ ] ] An object in CategoryOfColimitQuivers( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ) given by the above data Datum: [ [ [ 0 ], [ (P)-[(b)]->(L) ] ], [ ] ] Range: [ [ <(L)> ], [ ] ] An object in CategoryOfColimitQuivers( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ) given by the above data A morphism in CategoryOfColimitQuivers( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ) given by the above data gap> F := CreateBouquet( 3, [ 0, 0, 0, 2 ] ); <An object in FinBouquets> gap> Display( F ); ( { 0, 1, 2 }, { 0 ↦ 0, 1 ↦ 0, 2 ↦ 0, 3 ↦ 2 } ) gap> F_as_presheaf := ModelingObject( Chat, ModelingObject( FinBouquets, F ) ); <An object in PreSheaves( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ), SkeletalFinSets )> gap> Display( F_as_presheaf ); Image of <(P)>: { 0, 1, 2 } Image of <(L)>: { 0,..., 3 } Image of (P)-[(b)]->(L): { 0,..., 3 } ⱶ[ 0, 0, 0, 2 ]→ { 0, 1, 2 } An object in PreSheaves( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ), SkeletalFinSets ) given by the above data gap> F_as_coequalizer_object := CoYonedaLemmaOnObjects( F_as_presheaf ); <An object in FiniteColimitCompletionWithStrictCoproducts( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) )> gap> ColimitCompletionC := CapCategory( F_as_coequalizer_object ); FiniteColimitCompletionWithStrictCoproducts( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ) gap> Display( ColimitCompletionC ); A CAP category with name FiniteColimitCompletionWithStrictCoproducts( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ): 22 primitive operations were used to derive 64 operations for this category which algorithmically * IsCocartesianCategory and not yet algorithmically * IsFiniteCocompleteCategory gap> Display( F_as_coequalizer_object ); Image of <(V)>: [ 7, [ <(P)>, <(P)>, <(P)>, <(L)>, <(L)>, <(L)>, <(L)> ] ] An object in FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ) given by the above data Image of <(A)>: [ 4, [ <(P)>, <(P)>, <(P)>, <(P)> ] ] An object in FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ) given by the above data Image of (V)-[(s)]->(A): { 0,..., 3 } ⱶ[ 0, 0, 0, 2 ]→ { 0,..., 6 } [ (P)-[(P)]->(P), (P)-[(P)]->(P), (P)-[(P)]->(P), (P)-[(P)]->(P) ] A morphism in FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ) given by the above data Image of (V)-[(t)]->(A): { 0,..., 3 } ⱶ[ 3, 4, 5, 6 ]→ { 0,..., 6 } [ (P)-[(b)]->(L), (P)-[(b)]->(L), (P)-[(b)]->(L), (P)-[(b)]->(L) ] A morphism in FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ) given by the above data An object in PreSheaves( FreeCategory( RightQuiver( "q(V,A)[s:V->A,t:V->A]" ) ), FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ) ) given by the above data An object in FiniteColimitCompletionWithStrictCoproducts( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ) given by the above data gap> F_as_presheaf = > Coequalizer( AssociatedCoequalizerPairInPreSheaves( F_as_coequalizer_object )[2] ); true gap> F_as_colimit_quiver := AssociatedColimitQuiver( F_as_coequalizer_object ); <An object in CategoryOfColimitQuivers( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) )> gap> Display( F_as_colimit_quiver ); [ [ <(P)>, <(P)>, <(P)>, <(L)>, <(L)>, <(L)>, <(L)> ], [ [ 0, (P)-[(b)]->(L), 3 ], [ 0, (P)-[(b)]->(L), 4 ], [ 0, (P)-[(b)]->(L), 5 ], [ 2, (P)-[(b)]->(L), 6 ] ] ] An object in CategoryOfColimitQuivers( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ) given by the above data gap> F_as_presheaf = > Coequalizer( AssociatedCoequalizerPairInPreSheaves( F_as_colimit_quiver )[2] ); true
‣ CreateColimitQuiver( ColimitQuivers, list ) | ( operation ) |
Returns: a colimit quiver
‣ CreateMorphismOfColimitQuivers( source, list_of_images, target ) | ( operation ) |
Returns: a colimit quiver morphism
‣ CategoryOfPreSheavesOfUnderlyingCategory( ColimitQuiversC ) | ( attribute ) |
Returns: a CAP category
Given the category ColimitQuiversC=CategoryOfColimitQuivers( \(C\) ) of colimit quivers in a \(V\)-enriched category \(C\), return the associated category PreSheaves( \(C\), \(V\) ) of presheaves.
‣ IsCategoryOfColimitQuivers( category ) | ( filter ) |
Returns: true or false
The GAP category of categories of colimit quivers in a category.
‣ IsCellInCategoryOfColimitQuivers( cell ) | ( filter ) |
Returns: true or false
The GAP category of cells in the category of colimit quivers in a category.
‣ IsObjectInCategoryOfColimitQuivers( obj ) | ( filter ) |
Returns: true or false
The GAP category of objects in the category of colimit quivers in a category.
‣ IsMorphismInCategoryOfColimitQuivers( mor ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in the category of colimit quivers in a category.
generated by GAPDoc2HTML