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4 The elementary topos of finite bouquets
 4.1 Constructors
 4.2 Attributes
 4.3 Operations
 4.4 Global variables
 4.5 GAP Categories
 4.6 Example

4 The elementary topos of finite bouquets

4.1 Constructors

4.1-1 CategoryOfBouquetsEnrichedOver
‣ CategoryOfBouquetsEnrichedOver( B, C )( operation )

Returns: a CAP category

Construct the category of bouquets.

4.1-2 CreateBouquet
‣ CreateBouquet( arg1, arg2, arg3 )( operation )

4.1-3 CreateBouquet
‣ CreateBouquet( arg1, arg2 )( operation )

4.1-4 CreateBouquetMorphism
‣ CreateBouquetMorphism( arg1, arg2, arg3, arg4 )( operation )

4.1-5 Subobject
‣ Subobject( arg1, arg2, arg3 )( operation )

4.1-6 Subobject
‣ Subobject( arg1, arg2 )( operation )

4.2 Attributes

4.2-1 UnderlyingCategory
‣ UnderlyingCategory( fin_bouquets )( attribute )

4.2-2 YonedaEmbeddingOfUnderlyingCategory
‣ YonedaEmbeddingOfUnderlyingCategory( fin_bouquets )( attribute )

4.2-3 Loops
‣ Loops( bouquet )( attribute )

4.2-4 SvgString
‣ SvgString( cell )( attribute )

4.2-5 DotVertexLabelledDigraph
‣ DotVertexLabelledDigraph( cell )( operation )

4.3 Operations

4.4 Global variables

The quiver generating the category of bouquets

4.4-1 QuiverOfCategoryOfBouquets
‣ QuiverOfCategoryOfBouquets( global variable )

The category of bouquets as a category of presheaves with values in SkeletalFinSets.

4.4-2 FinBouquets
‣ FinBouquets( global variable )

4.5 GAP Categories

4.5-1 IsCategoryOfBouquets
‣ IsCategoryOfBouquets( category )( filter )

Returns: true or false

The GAP category of the category of bouquets.

4.5-2 IsCellInCategoryOfBouquets
‣ IsCellInCategoryOfBouquets( cell )( filter )

Returns: true or false

The GAP category of cells in the category of bouquets.

4.5-3 IsObjectInCategoryOfBouquets
‣ IsObjectInCategoryOfBouquets( obj )( filter )

Returns: true or false

The GAP category of objects in the category of bouquets.

4.5-4 IsMorphismInCategoryOfBouquets
‣ IsMorphismInCategoryOfBouquets( mor )( filter )

Returns: true or false

The GAP category of morphisms in the category of bouquets.

4.6 Example

In the following we construct the category of finite bouquets (or finite multisets or finite maps):

gap> FinBouquets;
CategoryOfBouquetsEnrichedOver( SkeletalFinSets )
gap> C := UnderlyingCategory( FinBouquets );
FreeCategory( RightQuiver( "q(V,L)[b:V->L]" ) )
gap> V := FinBouquets.V;
<A projective object in CategoryOfBouquetsEnrichedOver( SkeletalFinSets )>
gap> Display( V );
( { 0 }, { } )
gap> L := FinBouquets.L;
<A projective object in CategoryOfBouquetsEnrichedOver( SkeletalFinSets )>
gap> Display( L );
( { 0 }, { 0 ↦ 0 } )
gap> b := FinBouquets.b;
<A monomorphism in CategoryOfBouquetsEnrichedOver( SkeletalFinSets )>
gap> Display( b );
Image of <(V)>:
{ 0 } ⱶ[ 0 ]→ { 0 }

Image of <(L)>:
∅ ⱶ[  ]→ { 0 }

A morphism in CategoryOfBouquetsEnrichedOver( SkeletalFinSets )
given by the above data
gap> T := TerminalObject( FinBouquets );
<An object in CategoryOfBouquetsEnrichedOver( SkeletalFinSets )>
gap> Display( T );
( { 0 }, { 0 ↦ 0 } )
gap> T = L;
true
gap> M := CreateBouquet( 3, [ 1, 1, 1, 2 ] );
<An object in CategoryOfBouquetsEnrichedOver( SkeletalFinSets )>
gap> Display( M );
( { 0, 1, 2 }, { 0 ↦ 1, 1 ↦ 1, 2 ↦ 1, 3 ↦ 2 } )
gap> N := Colimit( [ V, V, V, L, L, L, L ],
>              [ [ 1, b, 3 ],
>                [ 1, b, 4 ],
>                [ 1, b, 5 ],
>                [ 2, b, 6 ] ] );
<An object in CategoryOfBouquetsEnrichedOver( SkeletalFinSets )>
gap> N = M;
true
gap> LL := Coproduct( L, L );
<An object in CategoryOfBouquetsEnrichedOver( SkeletalFinSets )>
gap> L2 := CreateBouquet( 1, [ 0, 0 ] );
<An object in CategoryOfBouquetsEnrichedOver( SkeletalFinSets )>
gap> epsilon := CreateBouquetMorphism( LL, [ 0, 0 ], [ 0, 1 ], L2 );
<A morphism in CategoryOfBouquetsEnrichedOver( SkeletalFinSets )>
gap> IsWellDefined( epsilon );
true
gap> IsEpimorphism( epsilon );
true
gap> IsSplitEpimorphism( epsilon );
false
gap> omega := SubobjectClassifier( FinBouquets );
<An object in CategoryOfBouquetsEnrichedOver( SkeletalFinSets )>
gap> Display( omega );
( { 0, 1 }, { 0 ↦ 0, 1 ↦ 1, 2 ↦ 1 } )
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