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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346#############################################################################
##
## CAP package
##
## Copyright 2015, Sebastian Gutsche, TU Kaiserslautern
## Sebastian Posur, RWTH Aachen
##
#! @Chapter Monoidal Categories
#!
##
#############################################################################
InstallValue( CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS, rec( ) );
InstallValue( MONOIDAL_CATEGORIES_METHOD_NAME_RECORD, rec(
## Basic Operations for Monoidal Categories
TensorProductOnObjects := rec(
installation_name := "TensorProductOnObjects",
filter_list := [ "object", "object" ],
cache_name := "TensorProductOnObjects",
return_type := "object" ),
TensorProductOnMorphismsWithGivenTensorProducts := rec(
installation_name := "TensorProductOnMorphismsWithGivenTensorProducts",
filter_list := [ "object", "morphism", "morphism", "object" ],
io_type := [ [ "s", "alpha", "beta", "r" ], [ "s", "r" ] ],
cache_name := "TensorProductOnMorphismsWithGivenTensorProducts",
return_type := "morphism" ),
AssociatorRightToLeftWithGivenTensorProducts := rec(
installation_name := "AssociatorRightToLeftWithGivenTensorProducts",
filter_list := [ "object", "object", "object", "object", "object" ],
io_type := [ [ "s", "a", "b", "c", "r" ], [ "s", "r" ] ],
cache_name := "AssociatorRightToLeftWithGivenTensorProducts",
return_type := "morphism" ),
AssociatorLeftToRightWithGivenTensorProducts := rec(
installation_name := "AssociatorLeftToRightWithGivenTensorProducts",
filter_list := [ "object", "object", "object", "object", "object" ],
io_type := [ [ "s", "a", "b", "c", "r" ], [ "s", "r" ] ],
cache_name := "AssociatorLeftToRightWithGivenTensorProducts",
return_type := "morphism" ),
TensorUnit := rec(
installation_name := "TensorUnit",
filter_list := [ "category" ],
cache_name := "TensorUnit",
return_type := "object" ),
LeftUnitorWithGivenTensorProduct := rec(
installation_name := "LeftUnitorWithGivenTensorProduct",
filter_list := [ "object", "object" ],
io_type := [ [ "a", "s" ], [ "s", "a" ] ],
cache_name := "LeftUnitorWithGivenTensorProduct",
return_type := "morphism" ),
LeftUnitorInverseWithGivenTensorProduct := rec(
installation_name := "LeftUnitorInverseWithGivenTensorProduct",
filter_list := [ "object", "object" ],
io_type := [ [ "a", "r" ], [ "a", "r" ] ],
cache_name := "LeftUnitorInverseWithGivenTensorProduct",
return_type := "morphism" ),
RightUnitorWithGivenTensorProduct := rec(
installation_name := "RightUnitorWithGivenTensorProduct",
filter_list := [ "object", "object" ],
io_type := [ [ "a", "s" ], [ "s", "a" ] ],
cache_name := "RightUnitorWithGivenTensorProduct",
return_type := "morphism" ),
RightUnitorInverseWithGivenTensorProduct := rec(
installation_name := "RightUnitorInverseWithGivenTensorProduct",
filter_list := [ "object", "object" ],
io_type := [ [ "a", "r" ], [ "a", "r" ] ],
cache_name := "RightUnitorInverseWithGivenTensorProduct",
return_type := "morphism" ),
BraidingWithGivenTensorProducts := rec(
installation_name := "BraidingWithGivenTensorProducts",
filter_list := [ "object", "object", "object", "object" ],
io_type := [ [ "s", "a", "b", "r" ], [ "s", "r" ] ],
cache_name := "BraidingWithGivenTensorProducts",
return_type := "morphism" ),
BraidingInverseWithGivenTensorProducts := rec(
installation_name := "BraidingInverseWithGivenTensorProducts",
filter_list := [ "object", "object", "object", "object" ],
io_type := [ [ "s", "a", "b", "r" ], [ "s", "r" ] ],
cache_name := "BraidingInverseWithGivenTensorProducts",
return_type := "morphism" ),
InternalHomOnObjects := rec(
installation_name := "InternalHomOnObjects",
filter_list := [ "object", "object" ],
io_type := [ [ "a", "b" ], [ "i" ] ],
cache_name := "InternalHomOnObjects",
return_type := "object" ),
InternalHomOnMorphismsWithGivenInternalHoms := rec(
installation_name := "InternalHomOnMorphismsWithGivenInternalHoms",
filter_list := [ "object", "morphism", "morphism", "object" ],
io_type := [ [ "s", "alpha", "beta", "r" ], [ "s", "r" ] ],
cache_name := "InternalHomOnMorphismsWithGivenInternalHoms",
return_type := "morphism" ),
EvaluationMorphismWithGivenSource := rec(
installation_name := "EvaluationMorphismWithGivenSource",
filter_list := [ "object", "object", "object" ],
io_type := [ [ "a", "b", "s" ], [ "s", "b" ] ],
cache_name := "EvaluationMorphismWithGivenSource",
return_type := "morphism" ),
CoevaluationMorphismWithGivenRange := rec(
installation_name := "CoevaluationMorphismWithGivenRange",
filter_list := [ "object", "object", "object" ],
io_type := [ [ "a", "b", "r" ], [ "a", "r" ] ],
cache_name := "CoevaluationMorphismWithGivenRange",
return_type := "morphism" ),
TensorProductToInternalHomAdjunctionMap := rec(
installation_name := "TensorProductToInternalHomAdjunctionMap",
filter_list := [ "object", "object", "morphism" ],
io_type := [ [ "a", "b", "f" ], [ "a", "i" ] ],
cache_name := "TensorProductToInternalHomAdjunctionMap",
return_type := "morphism",
no_with_given := true ),
InternalHomToTensorProductAdjunctionMap := rec(
installation_name := "InternalHomToTensorProductAdjunctionMap",
filter_list := [ "object", "object", "morphism" ],
io_type := [ [ "b", "c", "g" ], [ "t", "c" ] ],
cache_name := "InternalHomToTensorProductAdjunctionMap",
return_type := "morphism",
no_with_given := true ),
MonoidalPreComposeMorphismWithGivenObjects := rec(
installation_name := "MonoidalPreComposeMorphismWithGivenObjects",
filter_list := [ "object", "object", "object", "object", "object" ],
io_type := [ [ "s", "a", "b", "c", "r" ], [ "s", "r" ] ],
cache_name := "MonoidalPreComposeMorphismWithGivenObjects",
return_type := "morphism" ),
MonoidalPostComposeMorphismWithGivenObjects := rec(
installation_name := "MonoidalPostComposeMorphismWithGivenObjects",
filter_list := [ "object", "object", "object", "object", "object" ],
io_type := [ [ "s", "a", "b", "c", "r" ], [ "s", "r" ] ],
cache_name := "MonoidalPostComposeMorphismWithGivenObjects",
return_type := "morphism" ),
DualOnObjects := rec(
installation_name := "DualOnObjects",
filter_list := [ "object" ],
io_type := [ [ "a" ], [ "ad" ] ],
cache_name := "DualOnObjects",
return_type := "object" ),
DualOnMorphismsWithGivenDuals := rec(
installation_name := "DualOnMorphismsWithGivenDuals",
io_type := [ [ "s", "alpha", "r" ], [ "s", "r" ] ],
filter_list := [ "object", "morphism", "object" ],
cache_name := "DualOnMorphismsWithGivenDuals",
return_type := "morphism" ),
EvaluationForDualWithGivenTensorProduct := rec(
installation_name := "EvaluationForDualWithGivenTensorProduct",
filter_list := [ "object", "object", "object" ],
io_type := [ [ "s", "a", "r" ], [ "s", "r" ] ],
cache_name := "EvaluationForDualWithGivenTensorProduct",
return_type := "morphism" ),
CoevaluationForDualWithGivenTensorProduct := rec(
installation_name := "CoevaluationForDualWithGivenTensorProduct",
filter_list := [ "object", "object", "object" ],
io_type := [ [ "s", "a", "r" ], [ "s", "r" ] ],
cache_name := "CoevaluationForDualWithGivenTensorProduct",
return_type := "morphism" ),
MorphismToBidualWithGivenBidual := rec(
installation_name := "MorphismToBidualWithGivenBidual",
filter_list := [ "object", "object" ],
io_type := [ [ "a", "r" ], [ "a", "r" ] ],
cache_name := "MorphismToBidualWithGivenBidual",
return_type := "morphism" ),
MorphismFromBidualWithGivenBidual := rec(
installation_name := "MorphismFromBidualWithGivenBidual",
filter_list := [ "object", "object" ],
io_type := [ [ "a", "s" ], [ "s", "a" ] ],
cache_name := "MorphismFromBidualWithGivenBidual",
return_type := "morphism" ),
TensorProductInternalHomCompatibilityMorphismWithGivenObjects := rec(
installation_name := "TensorProductInternalHomCompatibilityMorphismWithGivenObjects",
filter_list := [ "object", "object", "object", "object", IsList ],
io_type := [ [ "a", "ap", "b", "bp", "L" ], [ "L_1", "L_2" ] ],
cache_name := "TensorProductInternalHomCompatibilityMorphismWithGivenObjects",
return_type := "morphism" ),
TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects := rec(
installation_name := "TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects",
filter_list := [ "object", "object", "object", "object", IsList ],
io_type := [ [ "a", "ap", "b", "bp", "L" ], [ "L_2", "L_1" ] ],
cache_name := "TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects",
return_type := "morphism" ),
TensorProductDualityCompatibilityMorphismWithGivenObjects := rec(
installation_name := "TensorProductDualityCompatibilityMorphismWithGivenObjects",
filter_list := [ "object", "object", "object", "object" ],
io_type := [ [ "s", "a", "b", "r" ], [ "s", "r" ] ],
cache_name := "TensorProductDualityCompatibilityMorphismWithGivenObjects",
return_type := "morphism" ),
MorphismFromTensorProductToInternalHomWithGivenObjects := rec(
installation_name := "MorphismFromTensorProductToInternalHomWithGivenObjects",
filter_list := [ "object", "object", "object", "object" ],
io_type := [ [ "s", "a", "b", "r" ], [ "s", "r" ] ],
cache_name := "MorphismFromTensorProductToInternalHomWithGivenObjects",
return_type := "morphism" ),
MorphismFromInternalHomToTensorProductWithGivenObjects := rec(
installation_name := "MorphismFromInternalHomToTensorProductWithGivenObjects",
filter_list := [ "object", "object", "object", "object" ],
io_type := [ [ "s", "a", "b", "r" ], [ "s", "r" ] ],
cache_name := "MorphismFromInternalHomToTensorProductWithGivenObjects",
return_type := "morphism" ),
TraceMap := rec(
installation_name := "TraceMap",
filter_list := [ "morphism" ],
io_type := [ [ "alpha" ], [ "u", "u" ] ],
cache_name := "TraceMap",
return_type := "morphism",
no_with_given := true ),
RankMorphism := rec(
installation_name := "RankMorphism",
filter_list := [ "object" ],
io_type := [ [ "a" ], [ "u", "u" ] ],
cache_name := "RankMorphism",
return_type := "morphism",
no_with_given := true ),
IsomorphismFromTensorProductToInternalHom := rec(
installation_name := "IsomorphismFromTensorProductToInternalHom",
filter_list := [ "object", "object" ],
io_type := [ [ "a", "b" ], [ "t", "i" ] ],
cache_name := "IsomorphismFromTensorProductToInternalHom",
return_type := "morphism",
no_with_given := true ),
IsomorphismFromInternalHomToTensorProduct := rec(
installation_name := "IsomorphismFromInternalHomToTensorProduct",
filter_list := [ "object", "object" ],
io_type := [ [ "a", "b" ], [ "i", "t" ] ],
cache_name := "IsomorphismFromInternalHomToTensorProduct",
return_type := "morphism",
no_with_given := true ),
IsomorphismFromInternalHomToDual := rec(
installation_name := "IsomorphismFromInternalHomToDual",
filter_list := [ "object" ],
io_type := [ [ "a" ], [ "i", "d" ] ],
cache_name := "IsomorphismFromInternalHomToDual",
return_type := "morphism",
no_with_given := true ),
IsomorphismFromDualToInternalHom := rec(
installation_name := "IsomorphismFromDualToInternalHom",
filter_list := [ "object" ],
io_type := [ [ "a" ], [ "d", "i" ] ],
cache_name := "IsomorphismFromDualToInternalHom",
return_type := "morphism",
no_with_given := true ),
UniversalPropertyOfDual := rec(
installation_name := "UniversalPropertyOfDual",
filter_list := [ "object", "object", "morphism" ],
io_type := [ [ "t", "a", "alpha" ], [ "t", "d" ] ],
cache_name := "UniversalPropertyOfDual",
return_type := "morphism",
no_with_given := true ),
LambdaIntroduction := rec(
installation_name := "LambdaIntroduction",
filter_list := [ "morphism" ],
io_type := [ [ "alpha" ], [ "u", "i" ] ],
cache_name := "LambdaIntroduction",
return_type := "morphism",
no_with_given := true ),
LambdaElimination := rec(
installation_name := "LambdaElimination",
filter_list := [ "object", "object", "morphism" ],
io_type := [ [ "a", "b", "alpha" ], [ "a", "b" ] ],
cache_name := "LambdaElimination",
return_type := "morphism",
no_with_given := true ),
IsomorphismFromObjectToInternalHomWithGivenInternalHom := rec(
installation_name := "IsomorphismFromObjectToInternalHomWithGivenInternalHom",
filter_list := [ "object", "object" ],
io_type := [ [ "a", "r" ], [ "a", "r" ] ],
cache_name := "IsomorphismFromObjectToInternalHomWithGivenInternalHom",
return_type := "morphism" ),
IsomorphismFromInternalHomToObjectWithGivenInternalHom := rec(
installation_name := "IsomorphismFromInternalHomToObjectWithGivenInternalHom",
filter_list := [ "object", "object" ],
io_type := [ [ "a", "s" ], [ "s", "a" ] ],
cache_name := "IsomorphismFromInternalHomToObjectWithGivenInternalHom",
return_type := "morphism" ),
LeftDistributivityExpandingWithGivenObjects := rec(
installation_name := "LeftDistributivityExpandingWithGivenObjects",
filter_list := [ "object", "object", IsList, "object" ],
io_type := [ [ "s", "a", "L", "r" ], [ "s", "r" ] ],
cache_name := "LeftDistributivityExpandingWithGivenObjects",
return_type := "morphism" ),
LeftDistributivityFactoringWithGivenObjects := rec(
installation_name := "LeftDistributivityFactoringWithGivenObjects",
filter_list := [ "object", "object", IsList, "object" ],
io_type := [ [ "s", "a", "L", "r" ], [ "s", "r" ] ],
cache_name := "LeftDistributivityFactoringWithGivenObjects",
return_type := "morphism" ),
RightDistributivityExpandingWithGivenObjects := rec(
installation_name := "RightDistributivityExpandingWithGivenObjects",
filter_list := [ "object", IsList, "object", "object" ],
io_type := [ [ "s", "L", "a", "r" ], [ "s", "r" ] ],
cache_name := "RightDistributivityExpandingWithGivenObjects",
return_type := "morphism" ),
RightDistributivityFactoringWithGivenObjects := rec(
installation_name := "RightDistributivityFactoringWithGivenObjects",
filter_list := [ "object", IsList, "object", "object" ],
io_type := [ [ "s", "L", "a", "r" ], [ "s", "r" ] ],
cache_name := "RightDistributivityFactoringWithGivenObjects",
return_type := "morphism" ),
) );
CAP_INTERNAL_ENHANCE_NAME_RECORD( MONOIDAL_CATEGORIES_METHOD_NAME_RECORD );
CAP_INTERNAL_INSTALL_ADDS_FROM_RECORD( MONOIDAL_CATEGORIES_METHOD_NAME_RECORD );
##
InstallMethod( AddTensorUnit,
[ IsCapCategory, IsFunction, IsInt ],
function( category, func, weight )
local wrapped_func;
wrapped_func := function( cat ) return func(); end;
AddTensorUnit( category, [ [ wrapped_func, [ ] ] ], weight );
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.TensorProductOnMorphisms :=
[ [ TensorProductOnMorphismsWithGivenTensorProducts, 1 ],
[ TensorProductOnObjects, 2 ] ];
##
InstallMethod( TensorProductOnMorphisms,
[ IsCapCategoryMorphism, IsCapCategoryMorphism ],
function( morphism_1, morphism_2 )
return TensorProductOnMorphismsWithGivenTensorProducts(
TensorProductOnObjects( Source( morphism_1 ), Source( morphism_2 ) ),
morphism_1, morphism_2,
TensorProductOnObjects( Range( morphism_1 ), Range( morphism_2 ) )
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.AssociatorRightToLeft :=
[ [ AssociatorRightToLeftWithGivenTensorProducts, 1 ],
[ TensorProductOnObjects, 4 ] ];
##
InstallMethod( AssociatorRightToLeft,
[ IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject ],
function( object_1, object_2, object_3 )
return AssociatorRightToLeftWithGivenTensorProducts(
TensorProductOnObjects( object_1, TensorProductOnObjects( object_2, object_3 ) ),
object_1, object_2, object_3,
TensorProductOnObjects( TensorProductOnObjects( object_1, object_2 ), object_3 )
);
end );
##
InstallMethod( AssociatorRightToLeft,
[ IsCapCategoryObject and IsCellOfSkeletalCategory, IsCapCategoryObject, IsCapCategoryObject ],
function( object_1, object_2, object_3 )
local source_and_range;
source_and_range := TensorProductOnObjects( object_1, TensorProductOnObjects( object_2, object_3 ) );
return AssociatorRightToLeftWithGivenTensorProducts(
source_and_range,
object_1, object_2, object_3,
source_and_range
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.AssociatorLeftToRight :=
[ [ AssociatorLeftToRightWithGivenTensorProducts, 1 ],
[ TensorProductOnObjects, 4 ] ];
##
InstallMethod( AssociatorLeftToRight,
[ IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject ],
function( object_1, object_2, object_3 )
return AssociatorLeftToRightWithGivenTensorProducts(
TensorProductOnObjects( TensorProductOnObjects( object_1, object_2 ), object_3 ),
object_1, object_2, object_3,
TensorProductOnObjects( object_1, TensorProductOnObjects( object_2, object_3 ) )
);
end );
##
InstallMethod( AssociatorLeftToRight,
[ IsCapCategoryObject and IsCellOfSkeletalCategory, IsCapCategoryObject, IsCapCategoryObject ],
function( object_1, object_2, object_3 )
local source_and_range;
source_and_range := TensorProductOnObjects( object_1, TensorProductOnObjects( object_2, object_3 ) );
return AssociatorLeftToRightWithGivenTensorProducts(
source_and_range,
object_1, object_2, object_3,
source_and_range
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.LeftUnitor :=
[ [ LeftUnitorWithGivenTensorProduct, 1 ],
[ TensorProductOnObjects, 1 ],
[ TensorUnit, 1 ] ];
##
InstallMethod( LeftUnitor,
[ IsCapCategoryObject ],
function( object )
local category;
category := CapCategory( object );
return LeftUnitorWithGivenTensorProduct( object, TensorProductOnObjects( TensorUnit( category ), object ) );
end );
##
InstallMethod( LeftUnitor,
[ IsCapCategoryObject and IsCellOfSkeletalCategory ],
function( object )
return LeftUnitorWithGivenTensorProduct( object, object );
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.LeftUnitorInverse :=
[ [ LeftUnitorInverseWithGivenTensorProduct, 1 ],
[ TensorProductOnObjects, 1 ],
[ TensorUnit, 1 ] ];
##
InstallMethod( LeftUnitorInverse,
[ IsCapCategoryObject ],
function( object )
local category;
category := CapCategory( object );
return LeftUnitorInverseWithGivenTensorProduct( object, TensorProductOnObjects( TensorUnit( category ), object ) );
end );
##
InstallMethod( LeftUnitorInverse,
[ IsCapCategoryObject and IsCellOfSkeletalCategory ],
function( object )
return LeftUnitorInverseWithGivenTensorProduct( object, object );
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.RightUnitor :=
[ [ RightUnitorWithGivenTensorProduct, 1 ],
[ TensorProductOnObjects, 1 ],
[ TensorUnit, 1 ] ];
##
InstallMethod( RightUnitor,
[ IsCapCategoryObject ],
function( object )
local category;
category := CapCategory( object );
return RightUnitorWithGivenTensorProduct( object, TensorProductOnObjects( object, TensorUnit( category ) ) );
end );
##
InstallMethod( RightUnitor,
[ IsCapCategoryObject and IsCellOfSkeletalCategory ],
function( object )
return RightUnitorWithGivenTensorProduct( object, object );
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.RightUnitorInverse :=
[ [ RightUnitorInverseWithGivenTensorProduct, 1 ],
[ TensorProductOnObjects, 1 ],
[ TensorUnit, 1 ] ];
##
InstallMethod( RightUnitorInverse,
[ IsCapCategoryObject ],
function( object )
local category;
category := CapCategory( object );
return RightUnitorInverseWithGivenTensorProduct( object, TensorProductOnObjects( object, TensorUnit( category ) ) );
end );
##
InstallMethod( RightUnitorInverse,
[ IsCapCategoryObject and IsCellOfSkeletalCategory ],
function( object )
return RightUnitorInverseWithGivenTensorProduct( object, object );
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.Braiding :=
[ [ BraidingWithGivenTensorProducts, 1 ],
[ TensorProductOnObjects, 2 ] ];
##
InstallMethod( Braiding,
[ IsCapCategoryObject, IsCapCategoryObject ],
function( object_1, object_2 )
return BraidingWithGivenTensorProducts( TensorProductOnObjects( object_1, object_2 ), object_1, object_2, TensorProductOnObjects( object_2, object_1 ) );
end );
##
InstallMethod( Braiding,
[ IsCapCategoryObject and IsCellOfSkeletalCategory, IsCapCategoryObject ],
function( object_1, object_2 )
local source_and_range;
source_and_range := TensorProductOnObjects( object_1, object_2 );
return BraidingWithGivenTensorProducts( source_and_range, object_1, object_2, source_and_range );
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.BraidingInverse :=
[ [ BraidingInverseWithGivenTensorProducts, 1 ],
[ TensorProductOnObjects, 2 ] ];
##
InstallMethod( BraidingInverse,
[ IsCapCategoryObject, IsCapCategoryObject ],
function( object_1, object_2 )
return BraidingInverseWithGivenTensorProducts( TensorProductOnObjects( object_2, object_1 ), object_1, object_2, TensorProductOnObjects( object_1, object_2 ) );
end );
##
InstallMethod( BraidingInverse,
[ IsCapCategoryObject and IsCellOfSkeletalCategory, IsCapCategoryObject ],
function( object_1, object_2 )
local source_and_range;
source_and_range := TensorProductOnObjects( object_1, object_2 );
return BraidingInverseWithGivenTensorProducts( source_and_range, object_1, object_2, source_and_range );
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.InternalHomOnMorphisms :=
[ [ InternalHomOnMorphismsWithGivenInternalHoms, 1 ],
[ InternalHomOnObjects, 2 ] ];
##
InstallMethod( InternalHomOnMorphisms,
[ IsCapCategoryMorphism, IsCapCategoryMorphism ],
function( morphism_1, morphism_2 )
return InternalHomOnMorphismsWithGivenInternalHoms(
InternalHomOnObjects( Range( morphism_1 ), Source( morphism_2 ) ),
morphism_1, morphism_2,
InternalHomOnObjects( Source( morphism_1 ), Range( morphism_2 ) )
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.EvaluationMorphism :=
[ [ EvaluationMorphismWithGivenSource, 1 ],
[ TensorProductOnObjects, 1 ],
[ InternalHomOnObjects, 1 ] ];
##
InstallMethod( EvaluationMorphism,
[ IsCapCategoryObject, IsCapCategoryObject ],
function( object_1, object_2 )
return EvaluationMorphismWithGivenSource(
object_1, object_2,
TensorProductOnObjects( InternalHomOnObjects( object_1, object_2 ), object_1 )
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.CoevaluationMorphism :=
[ [ CoevaluationMorphismWithGivenRange, 1 ],
[ TensorProductOnObjects, 1 ],
[ InternalHomOnObjects, 1 ] ];
##
InstallMethod( CoevaluationMorphism,
[ IsCapCategoryObject, IsCapCategoryObject ],
function( object_1, object_2 )
return CoevaluationMorphismWithGivenRange(
object_1, object_2,
InternalHomOnObjects( object_1, TensorProductOnObjects( object_2, object_1 ) )
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.MonoidalPreComposeMorphism :=
[ [ MonoidalPreComposeMorphismWithGivenObjects, 1 ],
[ TensorProductOnObjects, 1 ],
[ InternalHomOnObjects, 3 ] ];
##
InstallMethod( MonoidalPreComposeMorphism,
[ IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject ],
function( object_1, object_2, object_3 )
return MonoidalPreComposeMorphismWithGivenObjects(
TensorProductOnObjects( InternalHomOnObjects( object_1, object_2 ), InternalHomOnObjects( object_2, object_3 ) ),
object_1, object_2, object_3,
InternalHomOnObjects( object_1, object_3 )
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.MonoidalPostComposeMorphism :=
[ [ MonoidalPostComposeMorphismWithGivenObjects, 1 ],
[ TensorProductOnObjects, 1 ],
[ InternalHomOnObjects, 3 ] ];
##
InstallMethod( MonoidalPostComposeMorphism,
[ IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject ],
function( object_1, object_2, object_3 )
return MonoidalPostComposeMorphismWithGivenObjects(
TensorProductOnObjects( InternalHomOnObjects( object_2, object_3 ), InternalHomOnObjects( object_1, object_2 ) ),
object_1, object_2, object_3,
InternalHomOnObjects( object_1, object_3 )
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.DualOnMorphisms :=
[ [ DualOnMorphismsWithGivenDuals, 1 ],
[ DualOnObjects, 2 ] ];
##
InstallMethod( DualOnMorphisms,
[ IsCapCategoryMorphism ],
function( morphism )
return DualOnMorphismsWithGivenDuals(
DualOnObjects( Range( morphism ) ),
morphism,
DualOnObjects( Source( morphism ) )
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.EvaluationForDual :=
[ [ EvaluationForDualWithGivenTensorProduct, 1 ],
[ TensorProductOnObjects, 1 ],
[ DualOnObjects, 1 ],
[ TensorUnit, 1 ] ];
##
InstallMethod( EvaluationForDual,
[ IsCapCategoryObject ],
function( object )
local category;
category := CapCategory( object );
return EvaluationForDualWithGivenTensorProduct(
TensorProductOnObjects( DualOnObjects( object ), object ),
object,
TensorUnit( category )
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.CoevaluationForDual :=
[ [ CoevaluationForDualWithGivenTensorProduct, 1 ],
[ TensorProductOnObjects, 1 ],
[ DualOnObjects, 1 ],
[ TensorUnit, 1 ] ];
##
InstallMethod( CoevaluationForDual,
[ IsCapCategoryObject ],
function( object )
local category;
category := CapCategory( object );
return CoevaluationForDualWithGivenTensorProduct(
TensorUnit( category ),
object,
TensorProductOnObjects( object, DualOnObjects( object ) )
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.MorphismToBidual :=
[ [ MorphismToBidualWithGivenBidual, 1 ],
[ DualOnObjects, 2 ] ];
##
InstallMethod( MorphismToBidual,
[ IsCapCategoryObject ],
function( object )
return MorphismToBidualWithGivenBidual( object, DualOnObjects( DualOnObjects( object ) ) );
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.MorphismFromBidual :=
[ [ MorphismFromBidualWithGivenBidual, 1 ],
[ DualOnObjects, 2 ] ];
##
InstallMethod( MorphismFromBidual,
[ IsCapCategoryObject ],
function( object )
return MorphismFromBidualWithGivenBidual( object, DualOnObjects( DualOnObjects( object ) ) );
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.TensorProductInternalHomCompatibilityMorphism :=
[ [ TensorProductInternalHomCompatibilityMorphismWithGivenObjects, 1 ],
[ TensorProductOnObjects, 3 ],
[ InternalHomOnObjects, 3 ] ];
##
InstallMethod( TensorProductInternalHomCompatibilityMorphism,
[ IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject ],
function( object_1_1, object_1_2, object_2_1, object_2_2 )
return TensorProductInternalHomCompatibilityMorphismWithGivenObjects(
object_1_1, object_1_2, object_2_1, object_2_2,
[ TensorProductOnObjects( InternalHomOnObjects( object_1_1, object_1_2 ), InternalHomOnObjects( object_2_1, object_2_2 ) ),
InternalHomOnObjects( TensorProductOnObjects( object_1_1, object_2_1 ), TensorProductOnObjects( object_1_2, object_2_2 ) ) ]
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.TensorProductInternalHomCompatibilityMorphismInverse :=
[ [ TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects, 1 ],
[ TensorProductOnObjects, 3 ],
[ InternalHomOnObjects, 3 ] ];
##
InstallMethod( TensorProductInternalHomCompatibilityMorphismInverse,
[ IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject ],
function( object_1_1, object_1_2, object_2_1, object_2_2 )
return TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects(
object_1_1, object_1_2, object_2_1, object_2_2,
[ TensorProductOnObjects( InternalHomOnObjects( object_1_1, object_1_2 ), InternalHomOnObjects( object_2_1, object_2_2 ) ),
InternalHomOnObjects( TensorProductOnObjects( object_1_1, object_2_1 ), TensorProductOnObjects( object_1_2, object_2_2 ) ) ]
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.TensorProductDualityCompatibilityMorphism :=
[ [ TensorProductDualityCompatibilityMorphismWithGivenObjects, 1 ],
[ DualOnObjects, 3 ],
[ TensorProductOnObjects, 2 ] ];
##
InstallMethod( TensorProductDualityCompatibilityMorphism,
[ IsCapCategoryObject, IsCapCategoryObject ],
function( object_1, object_2 )
return TensorProductDualityCompatibilityMorphismWithGivenObjects(
DualOnObjects( TensorProductOnObjects( object_1, object_2 ) ),
object_1, object_2,
TensorProductOnObjects( DualOnObjects( object_1 ), DualOnObjects( object_2 ) )
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.MorphismFromTensorProductToInternalHom :=
[ [ MorphismFromTensorProductToInternalHomWithGivenObjects, 1 ],
[ DualOnObjects, 1 ],
[ TensorProductOnObjects, 1 ],
[ InternalHomOnObjects, 1 ] ];
##
InstallMethod( MorphismFromTensorProductToInternalHom,
[ IsCapCategoryObject, IsCapCategoryObject ],
function( object_1, object_2 )
return MorphismFromTensorProductToInternalHomWithGivenObjects(
TensorProductOnObjects( DualOnObjects( object_1 ), object_2 ),
object_1, object_2,
InternalHomOnObjects( object_1, object_2 )
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.MorphismFromInternalHomToTensorProduct :=
[ [ MorphismFromInternalHomToTensorProductWithGivenObjects, 1 ],
[ DualOnObjects, 1 ],
[ TensorProductOnObjects, 1 ],
[ InternalHomOnObjects, 1 ] ];
##
InstallMethod( MorphismFromInternalHomToTensorProduct,
[ IsCapCategoryObject, IsCapCategoryObject ],
function( object_1, object_2 )
return MorphismFromInternalHomToTensorProductWithGivenObjects(
InternalHomOnObjects( object_1, object_2 ),
object_1, object_2,
TensorProductOnObjects( DualOnObjects( object_1 ), object_2 )
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.IsomorphismFromInternalHomToObject :=
[ [ IsomorphismFromInternalHomToObjectWithGivenInternalHom, 1 ],
[ TensorUnit, 1 ],
[ InternalHomOnObjects, 1 ] ];
##
InstallMethod( IsomorphismFromInternalHomToObject,
[ IsCapCategoryObject ],
function( object )
local category;
category := CapCategory( object );
return IsomorphismFromInternalHomToObjectWithGivenInternalHom( object, InternalHomOnObjects( TensorUnit( category ), object ) );
end );
##
InstallMethod( IsomorphismFromInternalHomToObject,
[ IsCapCategoryObject and IsCellOfSkeletalCategory ],
function( object )
return IsomorphismFromInternalHomToObjectWithGivenInternalHom( object, object );
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.IsomorphismFromObjectToInternalHom :=
[ [ IsomorphismFromObjectToInternalHomWithGivenInternalHom, 1 ],
[ TensorUnit, 1 ],
[ InternalHomOnObjects, 1 ] ];
##
InstallMethod( IsomorphismFromObjectToInternalHom,
[ IsCapCategoryObject ],
function( object )
local category;
category := CapCategory( object );
return IsomorphismFromObjectToInternalHomWithGivenInternalHom( object, InternalHomOnObjects( TensorUnit( category ), object ) );
end );
##
InstallMethod( IsomorphismFromObjectToInternalHom,
[ IsCapCategoryObject and IsCellOfSkeletalCategory ],
function( object )
return IsomorphismFromObjectToInternalHomWithGivenInternalHom( object, object );
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.LeftDistributivityExpanding :=
[ [ TensorProductOnObjects, 1 + 2 ], ## 1 + Size( summands_list ) would be the correct number
[ DirectSum, 2 ] ];
##
InstallMethod( LeftDistributivityExpanding,
[ IsCapCategoryObject, IsList ],
function( object, summands_list )
return LeftDistributivityExpandingWithGivenObjects(
TensorProductOnObjects( object, DirectSum( summands_list ) ),
object, summands_list,
DirectSum( List( summands_list, summand -> TensorProductOnObjects( object, summand ) ) )
);
end );
##
InstallMethod( LeftDistributivityExpanding,
[ IsCapCategoryObject and IsCellOfSkeletalCategory, IsList ],
function( object, summands_list )
local source_and_range;
source_and_range := TensorProductOnObjects( object, DirectSum( summands_list ) );
return LeftDistributivityExpandingWithGivenObjects(
source_and_range,
object, summands_list,
source_and_range
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.LeftDistributivityFactoring :=
[ [ TensorProductOnObjects, 1 + 2 ], ## 1 + Size( summands_list ) would be the correct number
[ DirectSum, 2 ] ];
##
InstallMethod( LeftDistributivityFactoring,
[ IsCapCategoryObject, IsList ],
function( object, summands_list )
return LeftDistributivityFactoringWithGivenObjects(
DirectSum( List( summands_list, summand -> TensorProductOnObjects( object, summand ) ) ),
object, summands_list,
TensorProductOnObjects( object, DirectSum( summands_list ) )
);
end );
##
InstallMethod( LeftDistributivityFactoring,
[ IsCapCategoryObject and IsCellOfSkeletalCategory, IsList ],
function( object, summands_list )
local source_and_range;
source_and_range := TensorProductOnObjects( object, DirectSum( summands_list ) );
return LeftDistributivityFactoringWithGivenObjects(
source_and_range,
object, summands_list,
source_and_range
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.RightDistributivityExpanding :=
[ [ TensorProductOnObjects, 1 + 2 ], ## 1 + Size( summands_list ) would be the correct number
[ DirectSum, 2 ] ];
##
InstallMethod( RightDistributivityExpanding,
[ IsList, IsCapCategoryObject ],
function( summands_list, object )
return RightDistributivityExpandingWithGivenObjects(
TensorProductOnObjects( DirectSum( summands_list ), object ),
summands_list, object,
DirectSum( List( summands_list, summand -> TensorProductOnObjects( summand, object ) ) )
);
end );
##
InstallMethod( RightDistributivityExpanding,
[ IsList, IsCapCategoryObject and IsCellOfSkeletalCategory ],
function( summands_list, object )
local source_and_range;
source_and_range := TensorProductOnObjects( DirectSum( summands_list ), object );
return RightDistributivityExpandingWithGivenObjects(
source_and_range,
summands_list, object,
source_and_range
);
end );
##
CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.RightDistributivityFactoring :=
[ [ TensorProductOnObjects, 1 + 2 ], ## 1 + Size( summands_list ) would be the correct number
[ DirectSum, 2 ] ];
##
InstallMethod( RightDistributivityFactoring,
[ IsList, IsCapCategoryObject ],
function( summands_list, object )
return RightDistributivityFactoringWithGivenObjects(
DirectSum( List( summands_list, summand -> TensorProductOnObjects( summand, object ) ) ),
summands_list, object,
TensorProductOnObjects( DirectSum( summands_list ), object )
);
end );
##
InstallMethod( RightDistributivityFactoring,
[ IsList, IsCapCategoryObject and IsCellOfSkeletalCategory ],
function( summands_list, object )
local source_and_range;
source_and_range := TensorProductOnObjects( DirectSum( summands_list ), object );
return RightDistributivityFactoringWithGivenObjects(
source_and_range,
summands_list, object,
source_and_range
);
end );
InstallGlobalFunction( CAP_INTERNAL_FIND_APPEARANCE_OF_SYMBOL_IN_FUNCTION_FOR_MONOIDAL_CATEGORIES,
function( func, loop_multiplier )
local monoidal_symbols, symbol, list, added_symbol, added_symbols;
monoidal_symbols := CAP_INTERNAL_FIND_APPEARANCE_OF_SYMBOL_IN_FUNCTION( func, RecNames( CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS ), loop_multiplier );
list := [ ];
for symbol in monoidal_symbols do
added_symbols := CAP_INTERNAL_MONOIDAL_CATEGORIES_BASIC_OPERATIONS.( symbol[ 1 ] );
for added_symbol in added_symbols do
Add( list, [ added_symbol[ 1 ], added_symbol[ 2 ] * symbol[ 2 ] ] );
od;
od;
return list;
end );
##
InstallMethod( TensorProductOp,
[ IsList, IsCapCategoryObject ],
function( list, object )
local size, i;
size := Size( list );
if size = 1 then
return object;
else
for i in [ 2 .. size ] do
object := TensorProductOnObjects( object, list[i] );
od;
fi;
return object;
end );
##
##
InstallMethod( TensorProductOp,
[ IsList, IsCapCategoryMorphism ],
function( list, morphism )
local size, i;
size := Size( list );
if size = 1 then
return morphism;
else
for i in [ 2 .. size ] do
morphism := TensorProductOnMorphisms( morphism, list[i] );
od;
fi;
return morphism;
end );