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# SPDX-License-Identifier: GPL-2.0-or-later
# MonoidalCategories: Monoidal and monoidal (co)closed categories
#
# Implementations
#
##
InstallMethod( TestBraidingForInvertibility,
[ IsCapCategory, IsCapCategoryObject, IsCapCategoryObject ],
function( cat, object_1, object_2 )
local b12, bi12, b21, bi21, b12bi12, bi12b12, b21bi21, bi21b21;
Assert( 0, HasIsBraidedMonoidalCategory( cat ) and IsBraidedMonoidalCategory( cat ) );
Assert( 0, IsIdenticalObj( cat, CapCategory( object_1 ) ) );
Assert( 0, IsIdenticalObj( cat, CapCategory( object_2 ) ) );
b12 := Braiding( object_1, object_2 );
bi12 := BraidingInverse( object_1, object_2 );
b21 := Braiding( object_2, object_1 );
bi21 := BraidingInverse( object_2, object_1 );
Assert( 0, IsWellDefined( b12 ) );
Assert( 0, IsWellDefined( bi12 ) );
Assert( 0, IsWellDefined( b21 ) );
Assert( 0, IsWellDefined( bi21 ) );
b12bi12 := PreCompose( b12, bi12 );
bi12b12 := PreCompose( bi12, b12 );
b21bi21 := PreCompose( b21, bi21 );
bi21b21 := PreCompose( bi21, b21 );
Assert( 0, IsWellDefined( b12bi12 ) );
Assert( 0, IsWellDefined( bi12b12 ) );
Assert( 0, IsWellDefined( b21bi21 ) );
Assert( 0, IsWellDefined( bi21b21 ) );
return IsOne( b12bi12 ) and
IsOne( bi12b12 ) and
IsOne( b21bi21 ) and
IsOne( bi21b21 );
end );
##
InstallMethod( TestBraidingCompatibility,
[ IsCapCategory, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject ],
function( cat, object_1, object_2, object_3 )
local morphism_left, morphism_right;
Assert( 0, HasIsBraidedMonoidalCategory( cat ) and IsBraidedMonoidalCategory( cat ) );
Assert( 0, IsIdenticalObj( cat, CapCategory( object_1 ) ) );
Assert( 0, IsIdenticalObj( cat, CapCategory( object_2 ) ) );
Assert( 0, IsIdenticalObj( cat, CapCategory( object_3 ) ) );
morphism_left := Braiding( TensorProductOnObjects( cat, object_1, object_2 ), object_3 );
Assert( 0, IsWellDefined( morphism_left ) );
morphism_left := PreCompose( morphism_left, AssociatorRightToLeft( object_3, object_1, object_2 ) );
Assert( 0, IsWellDefined( morphism_left ) );
morphism_left := PreCompose( morphism_left,
TensorProductOnMorphisms( Braiding( object_3, object_1 ), IdentityMorphism( object_2 ) ) );
Assert( 0, IsWellDefined( morphism_left ) );
morphism_right := AssociatorLeftToRight( object_1, object_2, object_3 );
Assert( 0, IsWellDefined( morphism_right ) );
morphism_right := PreCompose( morphism_right,
TensorProductOnMorphisms( IdentityMorphism( object_1 ), Braiding( object_2, object_3 ) ) );
Assert( 0, IsWellDefined( morphism_right ) );
morphism_right := PreCompose( morphism_right, AssociatorRightToLeft( object_1, object_3, object_2 ) );
Assert( 0, IsWellDefined( morphism_right ) );
if not ( morphism_left = morphism_right ) then
return false;
fi;
morphism_left := Braiding( object_1, TensorProductOnObjects( cat, object_2, object_3 ) );
Assert( 0, IsWellDefined( morphism_left ) );
morphism_left := PreCompose( morphism_left, AssociatorLeftToRight( object_2, object_3, object_1 ) );
Assert( 0, IsWellDefined( morphism_left ) );
morphism_left := PreCompose( morphism_left,
TensorProductOnMorphisms( IdentityMorphism( object_2 ), Braiding( object_3, object_1 ) ) );
Assert( 0, IsWellDefined( morphism_left ) );
morphism_right := AssociatorRightToLeft( object_1, object_2, object_3 );
Assert( 0, IsWellDefined( morphism_right ) );
morphism_right := PreCompose( morphism_right,
TensorProductOnMorphisms( Braiding( object_1, object_2 ), IdentityMorphism( object_3 ) ) );
Assert( 0, IsWellDefined( morphism_right ) );
morphism_right := PreCompose( morphism_right, AssociatorLeftToRight( object_2, object_1, object_3 ) );
Assert( 0, IsWellDefined( morphism_right ) );
return morphism_left = morphism_right;
end );
##
InstallMethod( TestBraidingCompatibilityForAllTriplesInList,
[ IsCapCategory, IsList ],
function( cat, object_list )
local a, b, c, size, list, test;
size := Length( object_list );
list := [ 1 .. size ];
for a in list do
for b in list do
for c in list do
test := TestBraidingCompatibility( cat, object_list[a], object_list[b], object_list[c] );
if not test then
Print( "indices of failing triple: ", [ a, b, c ], "\n" );
return false;
fi;
od;
od;
od;
end );
##
InstallGlobalFunction( "BraidedMonoidalCategoriesTest",
function( cat, opposite, a, b )
local verbose,
a_op, braiding_a_b, braiding_a_b_op, braiding_inverse_a_b, braiding_inverse_a_b_op,
b_op, braiding_b_a, braiding_b_a_op, braiding_inverse_b_a, braiding_inverse_b_a_op;
a_op := Opposite( opposite, a );
b_op := Opposite( opposite, b );
verbose := ValueOption( "verbose" ) = true;
if IsEmpty( MissingOperationsForConstructivenessOfCategory( cat, "IsBraidedMonoidalCategory" ) ) then
Assert( 0, TestBraidingForInvertibility( cat, a, b ) );
Assert( 0, TestBraidingCompatibility( cat, a, b, a ) );
fi;
if IsEmpty( MissingOperationsForConstructivenessOfCategory( opposite, "IsBraidedMonoidalCategory" ) ) then
Assert( 0, TestBraidingForInvertibility( opposite, a_op, b_op ) );
Assert( 0, TestBraidingCompatibility( opposite, a_op, b_op, a_op ) );
fi;
if CanCompute( cat, "Braiding" ) then
if verbose then
# COVERAGE_IGNORE_NEXT_LINE
Display( "Testing 'Braiding' ..." );
fi;
braiding_a_b := Braiding( a, b );
braiding_b_a := Braiding( b, a );
braiding_inverse_a_b_op := BraidingInverse( opposite, a_op, b_op );
braiding_inverse_b_a_op := BraidingInverse( opposite, b_op, a_op );
Assert( 0, IsCongruentForMorphisms( braiding_inverse_a_b_op, Opposite( opposite, braiding_a_b ) ) );
Assert( 0, IsCongruentForMorphisms( braiding_inverse_b_a_op, Opposite( opposite, braiding_b_a ) ) );
fi;
if CanCompute( cat, "BraidingInverse" ) then
if verbose then
# COVERAGE_IGNORE_NEXT_LINE
Display( "Testing 'BraidingInverse' ..." );
fi;
braiding_inverse_a_b := BraidingInverse( a, b );
braiding_inverse_b_a := BraidingInverse( b, a );
braiding_a_b_op := Braiding( opposite, a_op, b_op );
braiding_b_a_op := Braiding( opposite, b_op, a_op );
Assert( 0, IsCongruentForMorphisms( braiding_a_b_op, Opposite( opposite, braiding_inverse_a_b ) ) );
Assert( 0, IsCongruentForMorphisms( braiding_b_a_op, Opposite( opposite, braiding_inverse_b_a ) ) );
fi;
end );
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