‣ IsSkeletalCategoryOfFiniteSets ( object ) | ( filter ) |
Returns: true
or false
The GAP category of a skeletal category of finite sets.
‣ IsObjectInSkeletalCategoryOfFiniteSets ( object ) | ( filter ) |
Returns: true
or false
The GAP category of objects in the skeletal category of finite sets.
‣ IsMorphismInSkeletalCategoryOfFiniteSets ( object ) | ( filter ) |
Returns: true
or false
The GAP category of morphisms in the skeletal category of finite sets.
‣ Length ( M ) | ( attribute ) |
Returns: an integer
The integer defining the skeletal finite set M, i.e., Length( FinSet( n ) ) = n
.
‣ AsList ( M ) | ( attribute ) |
Returns: a list
The list associated to a skeletal finite set, i.e., AsList( FinSet( n ) ) = [ 0 .. n - 1 ]
.
‣ AsList ( phi ) | ( attribute ) |
Returns: a list
The graph defining the skeletal finite set morphism phi, see MapOfFinSets
(2.3-5).
‣ SkeletalCategoryOfFiniteSets ( ) | ( operation ) |
Returns: a CAP category
Construct a category of skeletal finite sets.
‣ SkeletalFinSets | ( global variable ) |
The default instance of the category of skeletal finite sets. It is automatically created while loading this package.
‣ FinSet ( n ) | ( operation ) |
Returns: a CAP object
Construct a skeletal finite set residing in the default instance of the category of skeletal finite sets SkeletalFinSets
of order given by the nonnegative integer n.
gap> LoadPackage( "FinSetsForCAP", false ); true gap> m := FinSet( 7 ); |7| gap> Display( m ); { 0,..., 6 } gap> IsWellDefined( m ); true gap> String( m ); "FinSet( SkeletalFinSets, 7 )" gap> Display( List( m, x -> x^2 ) ); [ 0, 1, 4, 9, 16, 25, 36 ] gap> L := ObjectDatum( m ); 7 gap> mm := ObjectConstructor( SkeletalFinSets, BigInt( 7 ) ); |7| gap> m = mm; true gap> n := FinSet( -2 ); |-2| gap> IsWellDefined( n ); false gap> n := FinSet( 3 ); |3| gap> IsWellDefined( n ); true gap> p := FinSet( 4 ); |4| gap> IsWellDefined( p ); true gap> IsEqualForObjects( m, n ); false
‣ FinSet ( C, n ) | ( operation ) |
Returns: a CAP object
Construct a skeletal finite set residing in the given category of skeletal finite sets C of order given by the nonnegative integer n.
‣ MapOfFinSets ( s, G, t ) | ( operation ) |
Returns: a CAP morphism
Construct a map \phi:s\tot of the skeletal finite sets s and t, i.e., a morphism in the CAP category of s, where G is a list of integers in t describing the graph of \phi.
gap> LoadPackage( "FinSetsForCAP", false ); true gap> s := FinSet( 3 ); |3| gap> t := FinSet( 7 ); |7| gap> phi := MapOfFinSets( s, [ 6, 4, 4 ], t ); |3| → |7| gap> Display( AsList( phi ) ); [ 6, 4, 4 ] gap> IsWellDefined( phi ); true gap> Display( phi ); { 0, 1, 2 } ⱶ[ 6, 4, 4 ]→ { 0,..., 6 } gap> String( phi ); "MapOfFinSets( SkeletalFinSets, FinSet( SkeletalFinSets, 3 ), [ 6, 4, 4 ], Fin\ Set( SkeletalFinSets, 7 ) )" gap> s := ObjectConstructor( SkeletalFinSets, BigInt( 3 ) ); |3| gap> t := ObjectConstructor( SkeletalFinSets, BigInt( 7 ) ); |7| gap> phi := MorphismConstructor( > s, > [ BigInt( 6 ), BigInt( 4 ), BigInt( 4 ) ], > t > ); |3| → |7| gap> Display( MorphismDatum( phi ) ); [ 6, 4, 4 ] gap> IsWellDefined( phi ); true gap> Display( phi ); { 0, 1, 2 } ⱶ[ 6, 4, 4 ]→ { 0,..., 6 }
‣ ListOp ( s, f ) | ( operation ) |
Returns: a list
Returns List( AsList( s ), f )
.
‣ EmbeddingOfFinSets ( s, t ) | ( operation ) |
Returns: a CAP morphism
Construct the embedding \iota:s\tot of the finite sets s and t, where s must be subset of t.
‣ Preimage ( phi, t ) | ( operation ) |
Returns: a CAP object
Compute the Preimage of t under the morphism phi.
‣ ImageObject ( phi, s_ ) | ( operation ) |
Returns: a CAP object
Compute the image of s_ under the morphism phi.
‣ CallFuncList ( phi, L ) | ( operation ) |
Returns: a list
Returns the image of L[1]
under the map phi assuming L[1]
is a nonnegative integer smaller than Length( Source( phi ) )
.
‣ GeometricSum ( q, n ) | ( operation ) |
Returns: an integer
Returns (q^n-1)/(q-1) = Sum
( [ 0 .. n - 1 ], i -> q^i ), taking the corner case q= 1 into account.
‣ GeometricSumDiff1 ( q, n ) | ( operation ) |
Returns: an integer
Returns (1+((n-1)*q-n)*q^(n-1))/(q-1)^2 = Sum
( [ 0 .. n - 1 ], i -> i * q^(i - 1) ), taking the corner case q= 1 into account.
‣ RemIntWithDomain ( v, a, ab ) | ( operation ) |
Returns: an integer
Return the remainder of the natural number v less than ab modulo a, where ab is a multiple of a.
‣ QuoIntWithDomain ( v, a, ab ) | ( operation ) |
Returns: an integer
Return the quotient of the natural number v less than ab modulo a, where ab is a multiple of a.
‣ DivIntWithGivenQuotient ( a, d, q ) | ( operation ) |
Returns: an integer
Return the quotient q of the natural number a by the divior d.
‣ DigitInPositionalNotation ( v, i, l, b ) | ( operation ) |
Returns: an integer
Return the digit of index i in the b-adic expansion of length l of the natural number v.
gap> LoadPackage( "FinSetsForCAP", false ); true gap> L := FinSet( 0 ); |0| gap> M := FinSet( 2 ); |2| gap> N := FinSet( 3 ); |3| gap> IsHomSetInhabited( L, L ); true gap> IsHomSetInhabited( M, L ); false gap> IsHomSetInhabited( L, M ); true gap> IsHomSetInhabited( M, N ); true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> s := FinSet( 7 ); |7| gap> t := FinSet( 4 ); |4| gap> psi := MapOfFinSets( s, [ 0, 2, 1, 2, 1, 3 ], t ); |7| → |4| gap> IsWellDefined( psi ); false gap> psi := MapOfFinSets( s, [ 0, 2, 1, 2, 1, 3, -2 ], t ); |7| → |4| gap> IsWellDefined( psi ); false gap> psi := MapOfFinSets( s, [ 1, 2, 1, 4, 2, 1, 3 ], t ); |7| → |4| gap> IsWellDefined( psi ); false gap> psi := MapOfFinSets( s, [ 0, 2, 1, 3, 2, 1, 3 ], t ); |7| → |4| gap> IsWellDefined( psi ); true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> m := FinSet( 3 ); |3| gap> n := FinSet( 5 ); |5| gap> p := FinSet( 7 ); |7| gap> psi := MapOfFinSets( m, [ 1, 4, 2 ], n ); |3| → |5| gap> phi := MapOfFinSets( n, [ 0, 3, 5, 5, 2 ], p ); |5| → |7| gap> alpha := PreCompose( psi, phi ); |3| → |7| gap> Display( alpha ); { 0, 1, 2 } ⱶ[ 3, 2, 5 ]→ { 0,..., 6 }
gap> LoadPackage( "FinSetsForCAP", false ); true gap> m := FinSet( 3 ); |3| gap> n := FinSet( 5 ); |5| gap> p := FinSet( 7 ); |7| gap> psi := MapOfFinSets( m, [ 0, 2, 4 ], n ); |3| → |5| gap> IsEpimorphism( psi ); false gap> IsSplitEpimorphism( psi ); false gap> IsMonomorphism( psi ); true gap> IsSplitMonomorphism( psi ); true gap> psi; |3| ↪ |5| gap> psi := MapOfFinSets( p, [ 0, 2, 1, 2, 2, 1, 0 ], m ); |7| → |3| gap> IsEpimorphism( psi ); true gap> IsSplitEpimorphism( psi ); true gap> IsMonomorphism( psi ); false gap> IsSplitMonomorphism( psi ); false gap> psi; |7| ↠ |3| gap> id := IdentityMorphism( m ); |3| → |3| gap> IsIsomorphism( id ); true gap> id; |3| ⭇ |3|
gap> LoadPackage( "FinSetsForCAP", false ); true gap> m := FinSet( 8 ); |8| gap> IsInitial( m ); false gap> IsTerminal( m ); false gap> i := InitialObject( SkeletalFinSets ); |0| gap> Display( i ); ∅ gap> IsInitial( i ); true gap> IsTerminal( i ); false gap> iota := UniversalMorphismFromInitialObject( m ); |0| → |8| gap> Display( AsList( i ) ); [ ] gap> t := TerminalObject( SkeletalFinSets ); |1| gap> Display( AsList( t ) ); [ 0 ] gap> IsInitial( t ); false gap> IsTerminal( t ); true gap> pi := UniversalMorphismIntoTerminalObject( m ); |8| → |1| gap> IsIdenticalObj( Range( pi ), t ); true gap> pi_t := UniversalMorphismIntoTerminalObjectWithGivenTerminalObject( m, t ); |8| → |1| gap> Display( AsList( pi_t ) ); [ 0, 0, 0, 0, 0, 0, 0, 0 ] gap> pi = pi_t; true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> I := FinSet( 0 ); |0| gap> T := FinSet( 1 ); |1| gap> M := FinSet( 2 ); |2| gap> IsProjective( I ); true gap> IsProjective( T ); true gap> IsProjective( M ); true gap> IsOne( EpimorphismFromSomeProjectiveObject( I ) ); true gap> IsOne( EpimorphismFromSomeProjectiveObject( M ) ); true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> I := FinSet( 0 ); |0| gap> T := FinSet( 1 ); |1| gap> M := FinSet( 2 ); |2| gap> IsInjective( I ); false gap> IsInjective( T ); true gap> IsInjective( M ); true gap> IsIsomorphism( MonomorphismIntoSomeInjectiveObject( I ) ); false gap> IsMonomorphism( MonomorphismIntoSomeInjectiveObject( I ) ); true gap> IsOne( MonomorphismIntoSomeInjectiveObject( M ) ); true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> m := FinSet( 7 ); |7| gap> n := FinSet( 3 ); |3| gap> p := FinSet( 4 ); |4| gap> d := DirectProduct( [ m, n, p ] ); |84| gap> Display( AsList( d ) ); [ 0 .. 83 ] gap> pi1 := ProjectionInFactorOfDirectProduct( [ m, n, p ], 1 ); |84| → |7| gap> Display( pi1 ); { 0,..., 83 } ⱶ[ 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6,\ 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6,\ 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6,\ 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6,\ 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6,\ 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6 ]→ { 0,..., 6 } gap> pi2 := ProjectionInFactorOfDirectProduct( [ m, n, p ], 2 ); |84| → |3| gap> Display( pi2 ); { 0,..., 83 } ⱶ[ 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1,\ 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0,\ 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2,\ 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1,\ 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0,\ 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2 ]→ { 0, 1, 2 } gap> pi3 := ProjectionInFactorOfDirectProduct( [ m, n, p ], 3 ); |84| → |4| gap> Display( pi3 ); { 0,..., 83 } ⱶ[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\ 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1,\ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,\ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,\ 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3,\ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 ]→ { 0,..., 3 } gap> psi := UniversalMorphismIntoDirectProduct( [ m, n, p ], [ pi1, pi2, pi3 ] ); |84| → |84| gap> psi = IdentityMorphism( d ); true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> m := FinSet( 7 ); |7| gap> n := FinSet( 3 ); |3| gap> p := FinSet( 4 ); |4| gap> c := Coproduct( m, n, p ); |14| gap> Display( AsList( c ) ); [ 0 .. 13 ] gap> iota1 := InjectionOfCofactorOfCoproduct( [ m, n, p ], 1 ); |7| → |14| gap> IsWellDefined( iota1 ); true gap> Display( iota1 ); { 0,..., 6 } ⱶ[ 0 .. 6 ]→ { 0,..., 13 } gap> iota2 := InjectionOfCofactorOfCoproduct( [ m, n, p ], 2 ); |3| → |14| gap> IsWellDefined( iota2 ); true gap> Display( iota2 ); { 0, 1, 2 } ⱶ[ 7 .. 9 ]→ { 0,..., 13 } gap> iota3 := InjectionOfCofactorOfCoproduct( [ m, n, p ], 3 ); |4| → |14| gap> IsWellDefined( iota3 ); true gap> Display( iota3 ); { 0,..., 3 } ⱶ[ 10 .. 13 ]→ { 0,..., 13 } gap> psi := UniversalMorphismFromCoproduct( > [ m, n, p ], [ iota1, iota2, iota3 ] ); |14| → |14| gap> id := IdentityMorphism( Coproduct( [ m, n, p ] ) ); |14| → |14| gap> psi = id; true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> m := FinSet( 7 ); |7| gap> n := FinSet( 3 ); |3| gap> phi := MapOfFinSets( n, [ 6, 4, 4 ], m ); |3| → |7| gap> IsWellDefined( phi ); true gap> ImageObject( phi ); |2| gap> Length( ImageObject( phi ) ); 2 gap> s := FinSet( 2 ); |2| gap> I := ImageObject( phi, s ); |2| gap> Length( I ); 2
gap> LoadPackage( "FinSetsForCAP", false ); true gap> m := FinSet( 7 ); |7| gap> n := FinSet( 3 ); |3| gap> phi := MapOfFinSets( n, [ 6, 4, 4 ], m ); |3| → |7| gap> pi := ImageEmbedding( phi ); |2| → |7| gap> Display( pi ); { 0, 1 } ⱶ[ 4, 6 ]→ { 0,..., 6 }
gap> LoadPackage( "FinSetsForCAP", false ); true gap> m := FinSet( 7 ); |7| gap> n := FinSet( 3 ); |3| gap> phi := MapOfFinSets( n, [ 6, 4, 4 ], m ); |3| → |7| gap> IsWellDefined( phi ); true gap> f := CoastrictionToImage( phi ); |3| → |2| gap> Display( f ); { 0, 1, 2 } ⱶ[ 1, 0, 0 ]→ { 0, 1 } gap> IsWellDefined( f ); true gap> IsEpimorphism( f ); true gap> IsSplitEpimorphism( f ); true gap> m := FinSet( 77 ); |77| gap> n := FinSet( 4 ); |4| gap> phi := MapOfFinSets( n, [ 76, 1, 24, 1 ], m ); |4| → |77| gap> IsWellDefined( phi ); true gap> iota := ImageEmbedding( phi ); |3| → |77| gap> pi := CoastrictionToImage( phi ); |4| → |3| gap> Display( pi ); { 0,..., 3 } ⱶ[ 2, 0, 1, 0 ]→ { 0, 1, 2 } gap> IsWellDefined( pi ); true gap> IsEpimorphism( pi ); true gap> IsSplitEpimorphism( pi ); true gap> PreCompose( pi, iota ) = phi; true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> m := FinSet( 3 ); |3| gap> n := FinSet( 3 ); |3| gap> phi := MapOfFinSets( m, [ 1, 0, 1 ], n ); |3| → |3| gap> I := CoimageObject( phi ); |2| gap> IsMonomorphism( phi ); false gap> IsSplitMonomorphism( phi ); false gap> IsEpimorphism( phi ); false gap> IsSplitEpimorphism( phi ); false gap> pi := CoimageProjection( phi ); |3| → |2| gap> iota := AstrictionToCoimage( phi ); |2| → |3| gap> PreCompose( pi, iota ) = phi; true gap> Display( iota ); { 0, 1 } ⱶ[ 1, 0 ]→ { 0, 1, 2 } gap> Display( ImageEmbedding( phi ) ); { 0, 1 } ⱶ[ 0, 1 ]→ { 0, 1, 2 }
gap> LoadPackage( "FinSetsForCAP", false ); true gap> m := FinSet( 7 ); |7| gap> n := FinSet( 3 ); |3| gap> phi := MapOfFinSets( n, [ 6, 4, 4 ], m ); |3| → |7| gap> P := Preimage( phi, [ 2 ] );; gap> Display( P ); [ ] gap> P := Preimage( phi, AsList( FinSet( 5 ) ) );; gap> Display( P ); [ 1, 2 ]
gap> LoadPackage( "FinSetsForCAP", false ); true gap> S := FinSet( 5 ); |5| gap> T := FinSet( 3 ); |3| gap> f1 := MapOfFinSets( S, [ 2, 2, 0, 1, 1 ], T ); |5| → |3| gap> f2 := MapOfFinSets( S, [ 2, 1, 2, 0, 1 ], T ); |5| → |3| gap> f3 := MapOfFinSets( S, [ 2, 0, 1, 0, 1 ], T ); |5| → |3| gap> D := [ f1, f2, f3 ];; gap> Eq := Equalizer( D ); |2| gap> iota := EmbeddingOfEqualizer( D ); |2| → |5| gap> Display( iota ); { 0, 1 } ⱶ[ 0, 4 ]→ { 0,..., 4 } gap> phi := MapOfFinSets( FinSet( 2 ), [ 4, 0 ], S );; gap> IsWellDefined( phi ); true gap> psi := UniversalMorphismIntoEqualizer( D, phi ); |2| → |2| gap> IsWellDefined( psi ); true gap> Display( psi ); { 0, 1 } ⱶ[ 1, 0 ]→ { 0, 1 } gap> PreCompose( psi, iota ) = phi; true gap> D := [ f2, f3 ];; gap> Eq := Equalizer( D ); |3| gap> psi := EmbeddingOfEqualizer( D ); |3| → |5| gap> Display( psi ); { 0, 1, 2 } ⱶ[ 0, 3, 4 ]→ { 0,..., 4 }
gap> LoadPackage( "FinSetsForCAP", false ); true gap> m := FinSet( 5 ); |5| gap> n1 := FinSet( 3 ); |3| gap> iota1 := EmbeddingOfFinSets( n1, m ); |3| ↪ |5| gap> Display( iota1 ); { 0, 1, 2 } ⱶ[ 0 .. 2 ]→ { 0,..., 4 } gap> n2 := FinSet( 4 ); |4| gap> iota2 := EmbeddingOfFinSets( n2, m ); |4| ↪ |5| gap> Display( iota2 ); { 0,..., 3 } ⱶ[ 0 .. 3 ]→ { 0,..., 4 } gap> D := [ iota1, iota2 ];; gap> Fib := FiberProduct( D ); |3| gap> pi1 := ProjectionInFactorOfFiberProduct( D, 1 ); |3| → |3| gap> Display( pi1 ); { 0, 1, 2 } ⱶ[ 0, 1, 2 ]→ { 0, 1, 2 } gap> int1 := ImageObject( pi1 ); |3| gap> pi2 := ProjectionInFactorOfFiberProduct( D, 2 ); |3| → |4| gap> Display( pi2 ); { 0, 1, 2 } ⱶ[ 0, 1, 2 ]→ { 0,..., 3 } gap> int2 := ImageObject( pi2 ); |3| gap> omega1 := PreCompose( pi1, iota1 ); |3| → |5| gap> omega2 := PreCompose( pi2, iota2 ); |3| → |5| gap> omega1 = omega2; true gap> Display( omega1 ); { 0, 1, 2 } ⱶ[ 0, 1, 2 ]→ { 0,..., 4 }
gap> LoadPackage( "FinSetsForCAP", false ); true gap> s := FinSet( 5 ); |5| gap> t := FinSet( 4 ); |4| gap> f := MapOfFinSets( s, [ 2, 3, 3, 1, 3 ], t ); |5| → |4| gap> g := MapOfFinSets( s, [ 2, 2, 3, 1, 3 ], t ); |5| → |4| gap> D := [ f, g ];; gap> C := Coequalizer( D ); |3| gap> pi := ProjectionOntoCoequalizer( D ); |4| → |3| gap> Display( pi ); { 0,..., 3 } ⱶ[ 0, 1, 2, 2 ]→ { 0, 1, 2 } gap> tau := MapOfFinSets( t, [ 1, 0, 1, 1 ], FinSet( 2 ) ); |4| → |2| gap> phi := UniversalMorphismFromCoequalizer( D, tau ); |3| → |2| gap> Display( phi ); { 0, 1, 2 } ⱶ[ 1, 0, 1 ]→ { 0, 1 } gap> PreCompose( pi, phi ) = tau; true gap> s := FinSet( 5 ); |5| gap> t := FinSet( 4 ); |4| gap> f := MapOfFinSets( s, [ 1, 2, 2, 1, 3 ], t ); |5| → |4| gap> g := MapOfFinSets( s, [ 1, 2, 1, 1, 3 ], t ); |5| → |4| gap> D := [ f, g ];; gap> C := Coequalizer( D ); |3| gap> pi := ProjectionOntoCoequalizer( D ); |4| → |3| gap> Display( pi ); { 0,..., 3 } ⱶ[ 0, 1, 1, 2 ]→ { 0, 1, 2 } gap> PreCompose( f, pi ) = PreCompose( g, pi ); true gap> tau := MapOfFinSets( t, [ 0, 1, 1, 0 ], FinSet( 2 ) ); |4| → |2| gap> phi := UniversalMorphismFromCoequalizer( D, tau ); |3| → |2| gap> Display( phi ); { 0, 1, 2 } ⱶ[ 0, 1, 0 ]→ { 0, 1 } gap> PreCompose( pi, phi ) = tau; true gap> s := FinSet( 2 );; gap> t := FinSet( 3 );; gap> f := MapOfFinSets( s, [ 0, 1 ], t );; gap> IsWellDefined( f ); true gap> g := MapOfFinSets( s, [ 1, 2 ], t );; gap> IsWellDefined( g ); true gap> C := Coequalizer( [ f, g ] ); |1|
gap> LoadPackage( "FinSetsForCAP", false ); true gap> M := FinSet( 5 ); |5| gap> N1 := FinSet( 3 ); |3| gap> iota1 := EmbeddingOfFinSets( N1, M ); |3| ↪ |5| gap> Display( iota1 ); { 0, 1, 2 } ⱶ[ 0 .. 2 ]→ { 0,..., 4 } gap> N2 := FinSet( 2 ); |2| gap> iota2 := EmbeddingOfFinSets( N2, M ); |2| ↪ |5| gap> Display( iota2 ); { 0, 1 } ⱶ[ 0 .. 1 ]→ { 0,..., 4 } gap> D := [ iota1, iota2 ];; gap> Fib := FiberProduct( D ); |2| gap> pi1 := ProjectionInFactorOfFiberProduct( D, 1 ); |2| → |3| gap> Display( pi1 ); { 0, 1 } ⱶ[ 0, 1 ]→ { 0, 1, 2 } gap> pi2 := ProjectionInFactorOfFiberProduct( D, 2 ); |2| → |2| gap> Display( pi2 ); { 0, 1 } ⱶ[ 0, 1 ]→ { 0, 1 }
The easy way
gap> LoadPackage( "FinSetsForCAP", false ); true gap> D := [ pi1, pi2 ];; gap> UU := Pushout( D ); |3| gap> kappa1 := InjectionOfCofactorOfPushout( D, 1 ); |3| → |3| gap> Display( kappa1 ); { 0, 1, 2 } ⱶ[ 0, 1, 2 ]→ { 0, 1, 2 } gap> kappa2 := InjectionOfCofactorOfPushout( D, 2 ); |2| → |3| gap> Display( kappa2 ); { 0, 1 } ⱶ[ 0, 1 ]→ { 0, 1, 2 } gap> PreCompose( pi1, kappa1 ) = PreCompose( pi2, kappa2 ); true
The long way
gap> LoadPackage( "FinSetsForCAP", false ); true gap> Co := Coproduct( N1, N2 ); |5| gap> Display( Co ); { 0,..., 4 } gap> iota_1 := InjectionOfCofactorOfCoproduct( [ N1, N2 ], 1 ); |3| → |5| gap> Display( iota_1 ); { 0, 1, 2 } ⱶ[ 0 .. 2 ]→ { 0,..., 4 } gap> iota_2 := InjectionOfCofactorOfCoproduct( [ N1, N2 ], 2 ); |2| → |5| gap> Display( iota_2 ); { 0, 1 } ⱶ[ 3 .. 4 ]→ { 0,..., 4 } gap> alpha := PreCompose( pi1, iota_1 ); |2| → |5| gap> Display( alpha ); { 0, 1 } ⱶ[ 0, 1 ]→ { 0,..., 4 } gap> beta := PreCompose( pi2, iota_2 ); |2| → |5| gap> Display( beta ); { 0, 1 } ⱶ[ 3, 4 ]→ { 0,..., 4 } gap> Cq := Coequalizer( [ alpha, beta ] ); |3| gap> psi := ProjectionOntoCoequalizer( [ alpha, beta ] ); |5| → |3| gap> Display( psi ); { 0,..., 4 } ⱶ[ 0, 1, 2, 0, 1 ]→ { 0, 1, 2 } gap> Display( PreCompose( iota_1, psi ) ); { 0, 1, 2 } ⱶ[ 0, 1, 2 ]→ { 0, 1, 2 } gap> Display( PreCompose( iota_2, psi ) ); { 0, 1 } ⱶ[ 0, 1 ]→ { 0, 1, 2 } gap> PreCompose( alpha, psi ) = PreCompose( beta, psi ); true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> S := FinSet( 3 ); |3| gap> R := FinSet( 2 ); |2| gap> f := MapOfFinSets( S, [ 1, 1, 0 ], R ); |3| → |2| gap> IsWellDefined( f ); true gap> T := TerminalObject( SkeletalFinSets ); |1| gap> IsTerminal( T ); true gap> IsOne( CartesianLambdaIntroduction( IdentityMorphism( T ) ) ); true gap> lf := CartesianLambdaIntroduction( f ); |1| → |8| gap> Source( lf ) = T; true gap> Display( Range( lf ) ); { 0,..., 7 } gap> Display( lf ); { 0 } ⱶ[ 3 ]→ { 0,..., 7 } gap> elf := CartesianLambdaElimination( S, R, lf ); |3| → |2| gap> elf = f; true gap> L := MorphismsOfExternalHom( S, R );; gap> Display( L ); [ { 0, 1, 2 } ⱶ[ 0, 0, 0 ]→ { 0, 1 },\ { 0, 1, 2 } ⱶ[ 1, 0, 0 ]→ { 0, 1 },\ { 0, 1, 2 } ⱶ[ 0, 1, 0 ]→ { 0, 1 },\ { 0, 1, 2 } ⱶ[ 1, 1, 0 ]→ { 0, 1 },\ { 0, 1, 2 } ⱶ[ 0, 0, 1 ]→ { 0, 1 },\ { 0, 1, 2 } ⱶ[ 1, 0, 1 ]→ { 0, 1 },\ { 0, 1, 2 } ⱶ[ 0, 1, 1 ]→ { 0, 1 },\ { 0, 1, 2 } ⱶ[ 1, 1, 1 ]→ { 0, 1 } ] gap> Li := List( L, phi -> CartesianLambdaIntroduction( phi ) );; gap> Display( Li ); [ { 0 } ⱶ[ 0 ]→ { 0,..., 7 }, { 0 } ⱶ[ 1 ]→ { 0,..., 7 },\ { 0 } ⱶ[ 2 ]→ { 0,..., 7 }, { 0 } ⱶ[ 3 ]→ { 0,..., 7 },\ { 0 } ⱶ[ 4 ]→ { 0,..., 7 }, { 0 } ⱶ[ 5 ]→ { 0,..., 7 },\ { 0 } ⱶ[ 6 ]→ { 0,..., 7 }, { 0 } ⱶ[ 7 ]→ { 0,..., 7 } ] gap> List( L, phi -> > DirectProductToExponentialLeftAdjunctMorphism( T, S, phi ) ) = Li; true gap> List( Li, psi -> CartesianLambdaElimination( S, R, psi ) ) = L; true gap> List( Li, psi -> > ExponentialToDirectProductLeftAdjunctMorphism( S, R, psi ) ) = L; true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> m := FinSet( 3 ); |3| gap> n := FinSet( 4 ); |4| gap> f := MapOfFinSets( m, [ 1, 1, 0 ], m ); |3| → |3| gap> g := MapOfFinSets( n, [ 2, 1, 0, 1 ], m ); |4| → |3| gap> IsLiftable( f, g ); true gap> chi := Lift( f, g ); |3| → |4| gap> Display( chi ); { 0, 1, 2 } ⱶ[ 1, 1, 2 ]→ { 0,..., 3 } gap> PreCompose( Lift( f, g ), g ) = f; true gap> IsLiftable( g, f ); false gap> k := FinSet( 100000 ); |100000| gap> h := ListWithIdenticalEntries( Length( k ) - 3, 2 );; gap> h := Concatenation( h, [ 1, 0, 1 ] );; gap> h := MapOfFinSets( k, h, m ); |100000| → |3| gap> IsLiftable( f, h ); true gap> IsLiftable( h, f ); false
gap> LoadPackage( "FinSetsForCAP", false ); true gap> I := InitialObject( SkeletalFinSets ); |0| gap> iota := UniversalMorphismIntoTerminalObject( I ); |0| → |1| gap> id := IdentityMorphism( I ); |0| → |0| gap> IsColiftable( iota, id ); false gap> m := FinSet( 5 ); |5| gap> n := FinSet( 4 ); |4| gap> f := MapOfFinSets( m, [ 1, 1, 0, 0, 2 ], n ); |5| → |4| gap> g := MapOfFinSets( m, [ 4, 4, 3, 3, 4 ], m ); |5| → |5| gap> IsColiftable( f, g ); true gap> chi := Colift( f, g ); |4| → |5| gap> Display( chi ); { 0,..., 3 } ⱶ[ 3, 4, 4, 0 ]→ { 0,..., 4 } gap> PreCompose( f, Colift( f, g ) ) = g; true gap> IsColiftable( g, f ); false
gap> LoadPackage( "FinSetsForCAP" ); true gap> a := FinSet( 3 ); |3| gap> sa := SingletonMorphism( a );; gap> Display( sa ); { 0, 1, 2 } ⱶ[ 1, 2, 4 ]→ { 0,..., 7 } gap> sa = LowerSegmentOfRelation( a, a, CartesianDiagonal( a, 2 ) ); true gap> sa = UpperSegmentOfRelation( a, a, CartesianDiagonal( a, 2 ) ); true
gap> LoadPackage( "FinSetsForCAP", false ); true gap> T := TerminalObject( SkeletalFinSets );; gap> M := FinSet( 4 );; gap> N := FinSet( 3 );; gap> P := FinSet( 4 );; gap> G_f := [ 0, 1, 0, 2 ];; gap> f := MapOfFinSets( M, G_f, N );; gap> IsWellDefined( f ); true gap> G_g := [ 2, 2, 1, 0 ];; gap> g := MapOfFinSets( M, G_g, N );; gap> IsWellDefined( g ); true gap> DirectProduct( M, N );; gap> DirectProductOnMorphisms( f, g );; gap> CartesianAssociatorLeftToRight( M, N, P );; gap> CartesianAssociatorRightToLeft( M, N, P );; gap> TerminalObject( FinSets );; gap> CartesianLeftUnitor( M );; gap> CartesianLeftUnitorInverse( M );; gap> CartesianRightUnitor( M );; gap> CartesianRightUnitorInverse( M );; gap> CartesianBraiding( M, N );; gap> CartesianBraidingInverse( M, N );; gap> ExponentialOnObjects( M, N );; gap> ExponentialOnMorphisms( f, g );; gap> CartesianRightEvaluationMorphism( M, N );; gap> ExponentialToDirectProductRightAdjunctMorphism( M, N, > UniversalMorphismFromInitialObject( ExponentialOnObjects( M, N ) ) > );; gap> CartesianLeftEvaluationMorphism( M, N );; gap> CartesianLeftCoevaluationMorphism( N, M );; gap> CartesianLeftCoevaluationMorphism( T, T );; gap> DirectProductToExponentialLeftAdjunctMorphism( M, N, > UniversalMorphismIntoTerminalObject( DirectProduct( M, N ) ) > );; gap> ExponentialToDirectProductLeftAdjunctMorphism( M, N, > UniversalMorphismFromInitialObject( ExponentialOnObjects( M, N ) ) > );; gap> M := FinSet( 2 );; gap> N := FinSet( 2 );; gap> P := FinSet( 1 );; gap> Q := FinSet( 2 );; gap> CartesianPreComposeMorphism( M, N, P );; gap> CartesianPostComposeMorphism( M, N, P );; gap> DirectProductExponentialCompatibilityMorphism( [ M, N, P, Q ] );;
gap> LoadPackage( "FinSetsForCAP", false ); true gap> SubobjectClassifier( SkeletalFinSets ); |2| gap> Display( TruthMorphismOfFalse( SkeletalFinSets ) ); { 0 } ⱶ[ 0 ]→ { 0, 1 } gap> Display( TruthMorphismOfTrue( SkeletalFinSets ) ); { 0 } ⱶ[ 1 ]→ { 0, 1 } gap> Display( TruthMorphismOfNot( SkeletalFinSets ) ); { 0, 1 } ⱶ[ 1, 0 ]→ { 0, 1 } gap> CartesianSquareOfSubobjectClassifier( SkeletalFinSets ); |4| gap> Display( TruthMorphismOfAnd( SkeletalFinSets ) ); { 0,..., 3 } ⱶ[ 0, 0, 0, 1 ]→ { 0, 1 } gap> Display( TruthMorphismOfOr( SkeletalFinSets ) ); { 0,..., 3 } ⱶ[ 0, 1, 1, 1 ]→ { 0, 1 } gap> Display( TruthMorphismOfImplies( SkeletalFinSets ) ); { 0,..., 3 } ⱶ[ 1, 0, 1, 1 ]→ { 0, 1 } gap> S := FinSet( 5 ); |5| gap> A := FinSet( 2 ); |2| gap> m := MapOfFinSets( A, [ 0, 4 ], S ); |2| → |5| gap> Display( ClassifyingMorphismOfSubobject( m ) ); { 0,..., 4 } ⱶ[ 1, 0, 0, 0, 1 ]→ { 0, 1 }
gap> LoadPackage( "FinSetsForCAP", false ); true gap> M := FinSet( 7 ); |7| gap> N := FinSet( 3 ); |3| gap> iotaN := MapOfFinSets( N, [ 1, 2, 4 ], M ); |3| → |7| gap> NC := PseudoComplementSubobject( iotaN ); |4| gap> tauN := EmbeddingOfPseudoComplementSubobject( iotaN ); |4| ↪ |7| gap> Display( tauN ); { 0,..., 3 } ⱶ[ 0, 3, 5, 6 ]→ { 0,..., 6 } gap> L := FinSet( 4 ); |4| gap> iotaL := MapOfFinSets( L, [ 1, 3, 4, 6 ], M ); |4| → |7| gap> NIL := IntersectionSubobject( iotaN, iotaL ); |2| gap> iotaNiL := EmbeddingOfIntersectionSubobject( iotaN, iotaL ); |2| ↪ |7| gap> Display( iotaNiL ); { 0, 1 } ⱶ[ 1, 4 ]→ { 0,..., 6 } gap> NUL := UnionSubobject( iotaN, iotaL ); |5| gap> iotaNuL := EmbeddingOfUnionSubobject( iotaN, iotaL ); |5| → |7| gap> Display( iotaNuL ); { 0,..., 4 } ⱶ[ 1, 2, 3, 4, 6 ]→ { 0,..., 6 } gap> NPL := RelativePseudoComplementSubobject( iotaN, iotaL ); |6| gap> iotaNpL := EmbeddingOfRelativePseudoComplementSubobject( iotaN, iotaL ); |6| ↪ |7| gap> Display( iotaNpL ); { 0,..., 5 } ⱶ[ 0, 1, 3, 4, 5, 6 ]→ { 0,..., 6 }
Define two composable monos K \stackrel{l}{\hookrightarrow} L \stackrel{m}{\hookrightarrow} G in SkeletalFinSets
:
gap> LoadPackage( "FinSetsForCAP", false ); true gap> K := FinSet( 3 ); |3| gap> L := FinSet( 6 ); |6| gap> l := MapOfFinSets( K, [ 2 .. 4 ], L );; IsMonomorphism( l );; l; |3| ↪ |6| gap> Display( l ); { 0, 1, 2 } ⱶ[ 2 .. 4 ]→ { 0,..., 5 } gap> G := FinSet( 8 ); |8| gap> Display( G ); { 0,..., 7 } gap> m := MapOfFinSets( L, [ 0, 1, 2, 3, 5, 6 ], G ); |6| → |8| gap> Display( m ); { 0,..., 5 } ⱶ[ 0, 1, 2, 3, 5, 6 ]→ { 0,..., 7 }
Now we compute the pushout complement D \stackrel{c}{\hookrightarrow} G of K \stackrel{l}{\hookrightarrow} L \stackrel{m}{\hookrightarrow} G:
gap> LoadPackage( "FinSetsForCAP", false ); true gap> HasPushoutComplement( l, m ); true gap> c := PushoutComplement( l, m ); |5| → |8| gap> Display( c ); { 0,..., 4 } ⱶ[ 2, 3, 4, 5, 7 ]→ { 0,..., 7 }
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