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<p><a id="X7967FE8E7BBDF485" name="X7967FE8E7BBDF485"></a></p>
<div class="ChapSects"><a href="chap17_mj.html#X7967FE8E7BBDF485">17 <span class="Heading">Examples and Tests</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap17_mj.html#X7CA5B46383A5CE09">17.1 <span class="Heading">Dummy implementations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap17_mj.html#X82A623E78640E822">17.1-1 <span class="Heading">Dummy categories</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap17_mj.html#X7B19E3AE7AB1B31B">17.1-2 <span class="Heading">Dummy rings</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap17_mj.html#X7BAE6D0285667B1F">17.2 <span class="Heading">Finite skeletal discrete categories</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap17_mj.html#X78D1062D78BE08C1">17.3 <span class="Heading">Functors</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap17_mj.html#X7B339750826300B4">17.4 <span class="Heading">HandlePrecompiledTowers</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap17_mj.html#X7EA29B7D7BA60F42">17.5 <span class="Heading">Terminal category</span></a>
</span>
</div>
</div>
<h3>17 <span class="Heading">Examples and Tests</span></h3>
<p><a id="X7CA5B46383A5CE09" name="X7CA5B46383A5CE09"></a></p>
<h4>17.1 <span class="Heading">Dummy implementations</span></h4>
<p><a id="X82A623E78640E822" name="X82A623E78640E822"></a></p>
<h5>17.1-1 <span class="Heading">Dummy categories</span></h5>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "CAP", false );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">list_of_operations_to_install := [</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "ObjectConstructor",</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "MorphismConstructor",</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "ObjectDatum",</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "MorphismDatum",</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "IsCongruentForMorphisms",</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "PreCompose",</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "IdentityMorphism",</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "DirectSum",</span>
<span class="GAPprompt">></span> <span class="GAPinput">];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">dummy := DummyCategory( rec(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> list_of_operations_to_install := list_of_operations_to_install,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> properties := [ "IsAdditiveCategory" ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll( list_of_operations_to_install, o -> CanCompute( dummy, o ) );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAdditiveCategory( dummy );</span>
true
</pre></div>
<p><a id="X7B19E3AE7AB1B31B" name="X7B19E3AE7AB1B31B"></a></p>
<h5>17.1-2 <span class="Heading">Dummy rings</span></h5>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "CAP", false );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">DummyRing( );</span>
Dummy ring 1
<span class="GAPprompt">gap></span> <span class="GAPinput">DummyRing( );</span>
Dummy ring 2
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRing( DummyRing( ) );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">DummyCommutativeRing( );</span>
Dummy commutative ring 1
<span class="GAPprompt">gap></span> <span class="GAPinput">DummyCommutativeRing( );</span>
Dummy commutative ring 2
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRing( DummyCommutativeRing( ) );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCommutative( DummyCommutativeRing( ) );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">DummyField( );</span>
Dummy field 1
<span class="GAPprompt">gap></span> <span class="GAPinput">DummyField( );</span>
Dummy field 2
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRing( DummyField( ) );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsField( DummyField( ) );</span>
true
</pre></div>
<p><a id="X7BAE6D0285667B1F" name="X7BAE6D0285667B1F"></a></p>
<h4>17.2 <span class="Heading">Finite skeletal discrete categories</span></h4>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "CAP", false );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">D := FiniteSkeletalDiscreteCategory( [ 1 .. 5 ] );</span>
FiniteSkeletalDiscreteCategory( [ 1 .. 5 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">one := ObjectConstructor( D, 1 );</span>
<An object in FiniteSkeletalDiscreteCategory( [ 1 .. 5 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsWellDefinedForObjects( one );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">ObjectDatum( one ) = 1;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( one );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">IsEqualForObjects( one, D[1] );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">id_one := IdentityMorphism( D, one );</span>
<An identity morphism in FiniteSkeletalDiscreteCategory( [ 1 .. 5 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsWellDefinedForMorphisms( id_one );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">MorphismDatum( id_one ) = fail;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( id_one );</span>
1
|
| A morphism in FiniteSkeletalDiscreteCategory( [ 1 .. 5 ] )
v
1
<span class="GAPprompt">gap></span> <span class="GAPinput">IsEqualForMorphisms( PreCompose( id_one, id_one ), id_one );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( SetOfObjectsOfCategory( D ) );</span>
5
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( SetOfMorphismsOfFiniteCategory( D ) );</span>
5
</pre></div>
<p><a id="X78D1062D78BE08C1" name="X78D1062D78BE08C1"></a></p>
<h4>17.3 <span class="Heading">Functors</span></h4>
<p>We create a binary functor <span class="SimpleMath">\(F\)</span> with one covariant and one contravariant component in two ways. Here is the first way to model a binary functor:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ring := HomalgRingOfIntegers( );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">vec := LeftPresentations( ring );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F := CapFunctor( "CohomForVec", [ vec, [ vec, true ] ], vec );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">obj_func := function( A, B ) return TensorProductOnObjects( A, DualOnObjects( B ) ); end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mor_func := function( source, alpha, beta, range ) return TensorProductOnMorphismsWithGivenTensorProducts( source, alpha, DualOnMorphisms( beta ), range ); end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AddObjectFunction( F, obj_func );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AddMorphismFunction( F, mor_func );;</span>
</pre></div>
<p>CAP regards <span class="SimpleMath">\(F\)</span> as a binary functor on a technical level, as we can see by looking at its input signature:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">InputSignature( F );</span>
[ [ Category of left presentations of Z, false ], [ Category of left presentations of Z, true ] ]
</pre></div>
<p>We can see that <code class="code">ApplyFunctor</code> works both on two arguments and on one argument (in the product category).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V1 := TensorUnit( vec );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">V3 := DirectSum( V1, V1, V1 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pi1 := ProjectionInFactorOfDirectSum( [ V1, V1 ], 1 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pi2 := ProjectionInFactorOfDirectSum( [ V3, V1 ], 1 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">value1 := ApplyFunctor( F, pi1, pi2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">input := Product( pi1, Opposite( pi2 ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">value2 := ApplyFunctor( F, input );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCongruentForMorphisms( value1, value2 );</span>
true
</pre></div>
<p>Here is the second way to model a binary functor:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F2 := CapFunctor( "CohomForVec2", Product( vec, Opposite( vec ) ), vec );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AddObjectFunction( F2, a -> obj_func( a[1], Opposite( a[2] ) ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AddMorphismFunction( F2, function( source, datum, range ) return mor_func( source, datum[1], Opposite( datum[2] ), range ); end );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">value3 := ApplyFunctor( F2,input );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCongruentForMorphisms( value1, value3 );</span>
true
</pre></div>
<p>CAP regards <span class="SimpleMath">\(F2\)</span> as a unary functor on a technical level, as we can see by looking at its input signature:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">InputSignature( F2 );</span>
[ [ Product of: Category of left presentations of Z, Opposite( Category of left presentations of Z ), false ] ]
</pre></div>
<p>Installation of the first functor as a GAP-operation. It will be installed both as a unary and binary version.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">InstallFunctor( F, "F_installation" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F_installation( pi1, pi2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F_installation( input );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F_installationOnObjects( V1, V1 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F_installationOnObjects( Product( V1, Opposite( V1 ) ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F_installationOnMorphisms( pi1, pi2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F_installationOnMorphisms( input );;</span>
</pre></div>
<p>Installation of the second functor as a GAP-operation. It will be installed only as a unary version.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">InstallFunctor( F2, "F_installation2" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F_installation2( input );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F_installation2OnObjects( Product( V1, Opposite( V1 ) ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F_installation2OnMorphisms( input );;</span>
</pre></div>
<p><a id="X7B339750826300B4" name="X7B339750826300B4"></a></p>
<h4>17.4 <span class="Heading">HandlePrecompiledTowers</span></h4>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "CAP", false );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">dummy1 := CreateCapCategory( );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">dummy2 := CreateCapCategory( );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">dummy3 := CreateCapCategory( );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrintAndReturn := function ( string )</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Print( string, "\n" ); return string; end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">dummy1!.compiler_hints := rec( );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">dummy1!.compiler_hints.precompiled_towers := [</span>
<span class="GAPprompt">></span> <span class="GAPinput"> rec(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> remaining_constructors_in_tower := [ "Constructor1" ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> precompiled_functions_adder := cat -></span>
<span class="GAPprompt">></span> <span class="GAPinput"> PrintAndReturn( "Adding precompiled operations for Constructor1" ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> rec(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> remaining_constructors_in_tower := [ "Constructor1", "Constructor2" ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> precompiled_functions_adder := cat -></span>
<span class="GAPprompt">></span> <span class="GAPinput"> PrintAndReturn( "Adding precompiled operations for Constructor2" ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> ),</span>
<span class="GAPprompt">></span> <span class="GAPinput">];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">HandlePrecompiledTowers( dummy2, dummy1, "Constructor1" );</span>
Adding precompiled operations for Constructor1
<span class="GAPprompt">gap></span> <span class="GAPinput">HandlePrecompiledTowers( dummy3, dummy2, "Constructor2" );</span>
Adding precompiled operations for Constructor2
</pre></div>
<p><a id="X7EA29B7D7BA60F42" name="X7EA29B7D7BA60F42"></a></p>
<h4>17.5 <span class="Heading">Terminal category</span></h4>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "CAP", false );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">T := TerminalCategoryWithMultipleObjects( );</span>
TerminalCategoryWithMultipleObjects( )
<span class="GAPprompt">gap></span> <span class="GAPinput">i := InitialObject( T );</span>
<A zero object in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">t := TerminalObject( T );</span>
<A zero object in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">z := ZeroObject( T );</span>
<A zero object in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( i );</span>
InitialObject( )
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( t );</span>
TerminalObject( )
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( z );</span>
ZeroObject( )
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIdenticalObj( i, z );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIdenticalObj( t, z );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">id_z := IdentityMorphism( z );</span>
<A zero, identity morphism in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">fn_z := ZeroObjectFunctorial( T );</span>
<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsEqualForMorphisms( id_z, fn_z );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCongruentForMorphisms( id_z, fn_z );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">a := "a" / T;</span>
<A zero object in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( a );</span>
a
<span class="GAPprompt">gap></span> <span class="GAPinput">IsWellDefined( a );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">aa := ObjectConstructor( T, "a" );</span>
<A zero object in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( aa );</span>
a
<span class="GAPprompt">gap></span> <span class="GAPinput">IsEqualForObjects( a, aa );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIsomorphicForObjects( a, aa );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIsomorphism( SomeIsomorphismBetweenObjects( a, aa ) );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">b := "b" / T;</span>
<A zero object in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( b );</span>
b
<span class="GAPprompt">gap></span> <span class="GAPinput">IsEqualForObjects( a, b );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIsomorphicForObjects( a, b );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">mor_ab := SomeIsomorphismBetweenObjects( a, b );</span>
<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIsomorphism( mor_ab );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( mor_ab );</span>
a
|
| SomeIsomorphismBetweenObjects
v
b
<span class="GAPprompt">gap></span> <span class="GAPinput">Hom_ab := MorphismsOfExternalHom( a, b );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( Hom_ab );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">Hom_ab[1];</span>
<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Hom_ab[1] );</span>
a
|
| InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism
v
b
<span class="GAPprompt">gap></span> <span class="GAPinput">Hom_ab[1] = mor_ab;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">HomStructure( mor_ab );</span>
<A zero, identity morphism in TerminalCategoryWithSingleObject( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">m := MorphismConstructor( a, "m", b );</span>
<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( m );</span>
a
|
| m
v
b
<span class="GAPprompt">gap></span> <span class="GAPinput">IsWellDefined( m );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">n := MorphismConstructor( a, "n", b );</span>
<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( n );</span>
a
|
| n
v
b
<span class="GAPprompt">gap></span> <span class="GAPinput">IsEqualForMorphisms( m, n );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCongruentForMorphisms( m, n );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">m = n;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">hom_mn := HomStructure( m, n );</span>
<A zero, identity morphism in TerminalCategoryWithSingleObject( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">id := IdentityMorphism( a );</span>
<A zero, identity morphism in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( id );</span>
a
|
| IdentityMorphism( a )
v
a
<span class="GAPprompt">gap></span> <span class="GAPinput">m = id;</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">id = MorphismConstructor( a, "xyz", a );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">zero := ZeroMorphism( a, a );</span>
<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( zero );</span>
a
|
| ZeroMorphism( a, a )
v
a
<span class="GAPprompt">gap></span> <span class="GAPinput">id = zero;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLiftable( m, n );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">lift := Lift( m, n );</span>
<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( lift );</span>
a
|
| Lift( m, n )
v
a
<span class="GAPprompt">gap></span> <span class="GAPinput">IsColiftable( m, n );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">colift := Colift( m, n );</span>
<A zero, isomorphism in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( colift );</span>
b
|
| Colift( m, n )
v
b
<span class="GAPprompt">gap></span> <span class="GAPinput">DirectProduct( T, [ ] );</span>
<A zero object in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Equalizer( T, z, [ ] );</span>
<A zero object in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Coproduct( T, [ ] );</span>
<A zero object in TerminalCategoryWithMultipleObjects( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Coequalizer( T, z, [ ] );</span>
<A zero object in TerminalCategoryWithMultipleObjects( )>
</pre></div>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "CAP", false );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">T := TerminalCategoryWithSingleObject( );</span>
TerminalCategoryWithSingleObject( )
<span class="GAPprompt">gap></span> <span class="GAPinput">i := InitialObject( T );</span>
<A zero object in TerminalCategoryWithSingleObject( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">t := TerminalObject( T );</span>
<A zero object in TerminalCategoryWithSingleObject( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">z := ZeroObject( T );</span>
<A zero object in TerminalCategoryWithSingleObject( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( i );</span>
A zero object in TerminalCategoryWithSingleObject( ).
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( t );</span>
A zero object in TerminalCategoryWithSingleObject( ).
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( z );</span>
A zero object in TerminalCategoryWithSingleObject( ).
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIdenticalObj( i, z );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIdenticalObj( t, z );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsWellDefined( z );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( SetOfObjectsOfCategory( T ) );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">id_z := IdentityMorphism( z );</span>
<A zero, identity morphism in TerminalCategoryWithSingleObject( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">fn_z := ZeroObjectFunctorial( T );</span>
<A zero, identity morphism in TerminalCategoryWithSingleObject( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsWellDefined( fn_z );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsEqualForMorphisms( id_z, fn_z );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCongruentForMorphisms( id_z, fn_z );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( SetOfMorphismsOfFiniteCategory( T ) );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLiftable( id_z, fn_z );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Lift( id_z, fn_z );</span>
<A zero, identity morphism in TerminalCategoryWithSingleObject( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsColiftable( id_z, fn_z );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Colift( id_z, fn_z );</span>
<A zero, identity morphism in TerminalCategoryWithSingleObject( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">DirectProduct( T, [ ] );</span>
<A zero object in TerminalCategoryWithSingleObject( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Equalizer( T, z, [ ] );</span>
<A zero object in TerminalCategoryWithSingleObject( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Coproduct( T, [ ] );</span>
<A zero object in TerminalCategoryWithSingleObject( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Coequalizer( T, z, [ ] );</span>
<A zero object in TerminalCategoryWithSingleObject( )>
</pre></div>
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