In the following we construct the category of finite bouquets (or finite multisets or finite maps):
gap> FinBouquets;
FinBouquets
gap> C := UnderlyingCategory( FinBouquets );
FreeCategory( RightQuiver( "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 } )