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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( V )( attribute )

Returns: a CAP category

Construct the category of bouquets enriched over the category V.

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 EmbeddingOfUnderlyingCategory
‣ EmbeddingOfUnderlyingCategory( 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;
FinBouquets
gap> C := UnderlyingCategory( FinBouquets );
PathCategory( FinQuiver( "q(P,L)[b:P-≻L]" ) )
gap> P := FinBouquets.P;
<A projective object in FinBouquets>
gap> Display( P );
( { 0 }, { } )
gap> L := FinBouquets.L;
<A projective object in FinBouquets>
gap> Display( L );
( { 0 }, { 0 ↦ 0 } )
gap> b := FinBouquets.b;
<A monomorphism in FinBouquets>
gap> Display( b );
Image of <(P)>:
{ 0 } ⱶ[ 0 ]→ { 0 }

Image of <(L)>:
∅ ⱶ[  ]→ { 0 }

A morphism in FinBouquets
given by the above data
gap> T := TerminalObject( FinBouquets );
<An object in FinBouquets>
gap> Display( T );
( { 0 }, { 0 ↦ 0 } )
gap> T = L;
true
gap> M := CreateBouquet( 3, [ 1, 1, 1, 2 ] );
<An object in FinBouquets>
gap> Display( M );
( { 0, 1, 2 }, { 0 ↦ 1, 1 ↦ 1, 2 ↦ 1, 3 ↦ 2 } )
gap> N := Colimit( [ P, P, P, L, L, L, L ],
>              [ [ 1, b, 3 ],
>                [ 1, b, 4 ],
>                [ 1, b, 5 ],
>                [ 2, b, 6 ] ] );
<An object in FinBouquets>
gap> N = M;
true
gap> LL := Coproduct( L, L );
<An object in FinBouquets>
gap> L2 := CreateBouquet( 1, [ 0, 0 ] );
<An object in FinBouquets>
gap> epsilon := CreateBouquetMorphism( LL, [ 0, 0 ], [ 0, 1 ], L2 );
<A morphism in FinBouquets>
gap> IsWellDefined( epsilon );
true
gap> IsEpimorphism( epsilon );
true
gap> IsSplitEpimorphism( epsilon );
false
gap> omega := SubobjectClassifier( FinBouquets );
<An object in FinBouquets>
gap> Display( omega );
( { 0, 1 }, { 0 ↦ 0, 1 ↦ 1, 2 ↦ 1 } )
gap> PM := PowerObject( M );
<An object in FinBouquets>
gap> Display( PM );
( { 0,..., 7 },
  { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 2, 5 ↦ 3,
    6 ↦ 2, 7 ↦ 3, 8 ↦ 2, 9 ↦ 3, 10 ↦ 2, 11 ↦ 3,
    12 ↦ 2, 13 ↦ 3, 14 ↦ 2, 15 ↦ 3, 16 ↦ 2, 17 ↦ 3,
    18 ↦ 4, 19 ↦ 5, 20 ↦ 6, 21 ↦ 7, 22 ↦ 6, 23 ↦ 7,
    24 ↦ 6, 25 ↦ 7, 26 ↦ 6, 27 ↦ 7, 28 ↦ 6, 29 ↦ 7,
    30 ↦ 6, 31 ↦ 7, 32 ↦ 6, 33 ↦ 7, 34 ↦ 6, 35 ↦ 7,
    36 ↦ 4, 37 ↦ 5, 38 ↦ 6, 39 ↦ 7, 40 ↦ 6, 41 ↦ 7,
    42 ↦ 6, 43 ↦ 7, 44 ↦ 6, 45 ↦ 7, 46 ↦ 6, 47 ↦ 7,
    48 ↦ 6, 49 ↦ 7, 50 ↦ 6, 51 ↦ 7, 52 ↦ 6, 53 ↦ 7 } )
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