Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Jump to bottom

Choose hom(B,C) = C ⊗ B^v instead of B^v ⊗ C #1501

Draft
wants to merge 1 commit into
base: master
Choose a base branch
from
Draft

Choose hom(B,C) = C ⊗ B^v instead of B^v ⊗ C #1501

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
164 changes: 60 additions & 104 deletions MonoidalCategories/gap/RigidSymmetricClosedMonoidalCategoriesDerivedMethods.gi
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -49,11 +49,11 @@ AddDerivationToCAP( InternalHomOnMorphismsWithGivenInternalHoms,
# |
# |
# v
# a'^v ⊗ b
# b ⊗ a'^v
# |
# | Dual(alpha) ⊗ beta
# | beta ⊗ Dual(alpha)
# v
# a^v ⊗ b'
# b' ⊗ a^v
# |
# |
# v
Expand All @@ -64,7 +64,7 @@ AddDerivationToCAP( InternalHomOnMorphismsWithGivenInternalHoms,
return PreComposeList( cat,
internal_hom_source,
[ IsomorphismFromInternalHomToTensorProductWithDualObject( cat, Range( alpha ), Source( beta ) ),
TensorProductOnMorphisms( cat, dual_alpha, beta ),
TensorProductOnMorphisms( cat, beta, dual_alpha ),
IsomorphismFromTensorProductWithDualObjectToInternalHom( cat, Source( alpha ), Range( beta ) ) ],
internal_hom_range );

Expand Down Expand Up @@ -102,11 +102,10 @@ end : CategoryFilter := IsRigidSymmetricClosedMonoidalCategory );
AddDerivationToCAP( EvaluationMorphismWithGivenSource,
"EvaluationMorphismWithGivenSource using the rigidity of the monoidal category",
[ [ PreComposeList, 1 ],
[ TensorProductOnMorphisms, 3 ],
[ TensorProductOnMorphisms, 2 ],
[ IsomorphismFromInternalHomToTensorProductWithDualObject, 1 ],
[ IdentityMorphism, 3 ],
[ Braiding, 1 ],
[ DualOnObjects, 2 ],
[ IdentityMorphism, 2 ],
[ DualOnObjects, 1 ],
[ AssociatorLeftToRight, 1 ],
[ EvaluationForDual, 1 ],
[ RightUnitor, 1 ] ],
Expand All @@ -118,10 +117,6 @@ AddDerivationToCAP( EvaluationMorphismWithGivenSource,
# |
# | Isomorphism ⊗ id_a
# v
# (a^v ⊗ b) ⊗ a
# |
# | B_( Dual(a), b ) ⊗ id_a
# v
# (b ⊗ a^v) ⊗ a
# |
# | α_( ( b, a^v ), a )
Expand All @@ -142,9 +137,6 @@ AddDerivationToCAP( EvaluationMorphismWithGivenSource,
IsomorphismFromInternalHomToTensorProductWithDualObject( cat, a, b ),
IdentityMorphism( cat, a ) ),

TensorProductOnMorphisms( cat,
Braiding( cat, DualOnObjects( cat, a ), b ),
IdentityMorphism( cat, a ) ),
AssociatorLeftToRight( cat, b, DualOnObjects( cat, a ), a ),

TensorProductOnMorphisms( cat,
Expand All @@ -162,11 +154,9 @@ end : CategoryFilter := IsRigidSymmetricClosedMonoidalCategory );
AddDerivationToCAP( EvaluationMorphismWithGivenSource,
"EvaluationMorphismWithGivenSource using the rigidity and strictness of the monoidal category",
[ [ PreComposeList, 1 ],
[ TensorProductOnMorphisms, 3 ],
[ TensorProductOnMorphisms, 2 ],
[ IsomorphismFromInternalHomToTensorProductWithDualObject, 1 ],
[ IdentityMorphism, 3 ],
[ Braiding, 1 ],
[ DualOnObjects, 1 ],
[ IdentityMorphism, 2 ],
[ EvaluationForDual, 1 ] ],

function( cat, a, b, internal_hom_tensored_a )
Expand All @@ -176,10 +166,6 @@ AddDerivationToCAP( EvaluationMorphismWithGivenSource,
# |
# | Isomorphism ⊗ id_a
# v
# a^v ⊗ b ⊗ a
# |
# | B_( Dual(a), b ) ⊗ id_a
# v
# b ⊗ a^v ⊗ a
# |
# | id_b ⊗ ev_a
Expand All @@ -192,10 +178,6 @@ AddDerivationToCAP( EvaluationMorphismWithGivenSource,
IsomorphismFromInternalHomToTensorProductWithDualObject( cat, a, b ),
IdentityMorphism( cat, a ) ),

TensorProductOnMorphisms( cat,
Braiding( cat, DualOnObjects( cat, a ), b ),
IdentityMorphism( cat, a ) ),

TensorProductOnMorphisms( cat,
IdentityMorphism( cat, b ),
EvaluationForDual( cat, a ) ) ],
Expand All @@ -209,13 +191,12 @@ end : CategoryFilter := cat -> HasIsRigidSymmetricClosedMonoidalCategory( cat )
AddDerivationToCAP( CoevaluationMorphismWithGivenRange,
"CoevaluationMorphismWithGivenRange using the rigidity of the monoidal category",
[ [ DualOnObjects, 1 ],
[ IdentityMorphism, 2 ],
[ IdentityMorphism, 1 ],
[ PreComposeList, 1 ],
[ LeftUnitorInverse, 1 ],
[ TensorProductOnMorphisms, 3 ],
[ RightUnitorInverse, 1 ],
[ TensorProductOnMorphisms, 1 ],
[ CoevaluationForDual, 1 ],
[ Braiding, 2 ],
[ AssociatorLeftToRight, 1 ],
[ AssociatorRightToLeft, 1 ],
[ IsomorphismFromTensorProductWithDualObjectToInternalHom, 1 ],
[ TensorProductOnObjects, 1 ] ],

Expand All @@ -224,25 +205,17 @@ AddDerivationToCAP( CoevaluationMorphismWithGivenRange,

# a
# |
# | (λ_a)^-1
# v
# 1 ⊗ a
# |
# | coev_b ⊗ id_a
# v
# (b ⊗ b^v) ⊗ a
# |
# | B_( b, b^v ) ⊗ id_a
# | (ρ_a)^-1
# v
# (b^v ⊗ b) ⊗ a
# a ⊗ 1
# |
# | α_( ( b^v, b ), a )
# | id_a ⊗ coev_b
# v
# b^v ⊗ (b ⊗ a)
# a ⊗ (b ⊗ b^v)
# |
# | id_(b^v) ⊗ B_( b, a )
# | α_( a, ( b, b^v ) )
# v
# b^v ⊗ (a ⊗ b)
# (a ⊗ b) ⊗ b^v
# |
# | Isomorphism
# v
Expand All @@ -254,21 +227,13 @@ AddDerivationToCAP( CoevaluationMorphismWithGivenRange,

morphism := PreComposeList( cat,
a,
[ LeftUnitorInverse( cat, a ),

TensorProductOnMorphisms( cat,
CoevaluationForDual( cat, b ),
id_a ),
[ RightUnitorInverse( cat, a ),

TensorProductOnMorphisms( cat,
Braiding( cat, b, dual_b ),
id_a ),

AssociatorLeftToRight( cat, dual_b, b, a ),
id_a,
CoevaluationForDual( cat, b ) ),

TensorProductOnMorphisms( cat,
IdentityMorphism( cat, dual_b ),
Braiding( cat, b, a ) ),
AssociatorRightToLeft( cat, a, b, dual_b ),

IsomorphismFromTensorProductWithDualObjectToInternalHom( cat,
b,
Expand All @@ -283,33 +248,24 @@ end : CategoryFilter := IsRigidSymmetricClosedMonoidalCategory );
AddDerivationToCAP( CoevaluationMorphismWithGivenRange,
"CoevaluationMorphismWithGivenRange using the rigidity of the monoidal category",
[ [ DualOnObjects, 1 ],
[ IdentityMorphism, 2 ],
[ IdentityMorphism, 1 ],
[ PreComposeList, 1 ],
[ TensorProductOnMorphisms, 3 ],
[ TensorProductOnMorphisms, 1 ],
[ CoevaluationForDual, 1 ],
[ Braiding, 2 ],
[ IsomorphismFromTensorProductWithDualObjectToInternalHom, 1 ],
[ TensorProductOnObjects, 1 ] ],

function( cat, a, b, internal_hom )
local dual_b, id_a, morphism;

# 1 ⊗ a
# |
# | coev_b ⊗ id_a
# v
# b ⊗ b^v ⊗ a
# |
# | B_( b, b^v ) ⊗ id_a
# v
# b^v ⊗ b ⊗ a
# |
# | id_(b^v) ⊗ B_( b, a )
# v
# b^v ⊗ a ⊗ b
# |
# | Isomorphism
# v
# a
# |
# | id_a ⊗ coev_b
# v
# (a ⊗ b) ⊗ b^v
# |
# | Isomorphism
# v
# Hom(b, a ⊗ b)

dual_b := DualOnObjects( cat, b );
Expand All @@ -319,16 +275,8 @@ AddDerivationToCAP( CoevaluationMorphismWithGivenRange,
morphism := PreComposeList( cat,
a,
[ TensorProductOnMorphisms( cat,
CoevaluationForDual( cat, b ),
id_a ),

TensorProductOnMorphisms( cat,
Braiding( cat, b, dual_b ),
id_a ),

TensorProductOnMorphisms( cat,
IdentityMorphism( cat, dual_b ),
Braiding( cat, b, a ) ),
id_a,
CoevaluationForDual( cat, b ) ),

IsomorphismFromTensorProductWithDualObjectToInternalHom( cat,
b,
Expand Down Expand Up @@ -388,6 +336,8 @@ AddDerivationToCAP( MorphismFromInternalHomToTensorProductWithGivenObjects,
"MorphismFromInternalHomToTensorProductWithGivenObjects using TensorProductInternalHomCompatibilityMorphismInverse",
[ [ TensorUnit, 1 ],
[ PostComposeList, 1 ],
[ Braiding, 1 ],
[ DualOnObjects, 1 ],
[ TensorProductOnMorphisms, 1 ],
[ IsomorphismFromInternalHomIntoTensorUnitToDualObject, 1 ],
[ IsomorphismFromInternalHomToObject, 1 ],
Expand All @@ -401,6 +351,10 @@ AddDerivationToCAP( MorphismFromInternalHomToTensorProductWithGivenObjects,

# inverse of the derivation of MorphismFromTensorProductToInternalHomWithGivenObjects using TensorProductInternalHomCompatibilityMorphism

# b ⊗ a^v
# ʌ
# | B_( a^v, b )
# |
# a^v ⊗ b
# ʌ
# |
Expand All @@ -417,6 +371,8 @@ AddDerivationToCAP( MorphismFromInternalHomToTensorProductWithGivenObjects,
unit := TensorUnit( cat );

return PostComposeList( cat, [
Braiding( cat, DualOnObjects( cat, a ), b ),

TensorProductOnMorphisms( cat,
IsomorphismFromInternalHomIntoTensorUnitToDualObject( cat, a ),
IsomorphismFromInternalHomToObject( cat, b ) ),
Expand All @@ -437,9 +393,7 @@ AddDerivationToCAP( CoevaluationForDualWithGivenTensorProduct,
[ [ IdentityMorphism, 1 ],
[ PreComposeList, 1 ],
[ LambdaIntroduction, 1 ],
[ IsomorphismFromInternalHomToTensorProductWithDualObject, 1 ],
[ Braiding, 1 ],
[ DualOnObjects, 1 ] ],
[ IsomorphismFromInternalHomToTensorProductWithDualObject, 1 ] ],

function( cat, unit, a, tensor_object )
local morphism;
Expand All @@ -452,19 +406,14 @@ AddDerivationToCAP( CoevaluationForDualWithGivenTensorProduct,
# |
# | Isomorphism
# v
# a^v ⊗ a
# |
# | B_( a^v, a )
# v
# a ⊗ a^v

morphism := IdentityMorphism( cat, a );

morphism := PreComposeList( cat,
unit,
[ LambdaIntroduction( cat, morphism ),
IsomorphismFromInternalHomToTensorProductWithDualObject( cat, a, a ),
Braiding( cat, DualOnObjects( cat, a ), a ) ],
IsomorphismFromInternalHomToTensorProductWithDualObject( cat, a, a ) ],
tensor_object );

return morphism;
Expand All @@ -478,6 +427,8 @@ AddDerivationToCAP( TraceMap,
[ PreComposeList, 1 ],
[ LambdaIntroduction, 1 ],
[ IsomorphismFromInternalHomToTensorProductWithDualObject, 1 ],
[ Braiding, 1 ],
[ DualOnObjects, 1 ],
[ EvaluationForDual, 1 ] ],

function( cat, alpha )
Expand All @@ -493,6 +444,10 @@ AddDerivationToCAP( TraceMap,
# |
# | Isomorphism
# v
# a ⊗ a^v
# |
# | B_( a, a^v )
# v
# a^v ⊗ a
# |
# | ev_a
Expand All @@ -507,6 +462,7 @@ AddDerivationToCAP( TraceMap,
unit,
[ LambdaIntroduction( cat, alpha ),
IsomorphismFromInternalHomToTensorProductWithDualObject( cat, a, a ),
Braiding( cat, a, DualOnObjects( a ) ),
EvaluationForDual( cat, a ) ],
unit );

Expand Down Expand Up @@ -612,8 +568,8 @@ end : CategoryFilter := IsRigidSymmetricClosedMonoidalCategory );
AddFinalDerivationBundle( "deriving the internal hom by tensoring with the dual object",
[ [ IdentityMorphism, 1 ],
[ DualOnObjects, 1 ],
[ RightUnitor, 1 ],
[ RightUnitorInverse, 1 ],
[ LeftUnitor, 1 ],
[ LeftUnitorInverse, 1 ],
[ TensorProductOnObjects, 1 ] ],
[ InternalHomOnObjects,
InternalHomOnMorphismsWithGivenInternalHoms,
Expand All @@ -639,7 +595,7 @@ AddFinalDerivationBundle( "deriving the internal hom by tensoring with the dual
[ DualOnObjects, 1 ] ],
function( cat, a, b )

return IdentityMorphism( cat, TensorProductOnObjects( cat, DualOnObjects( cat, a ), b ) );
return IdentityMorphism( cat, TensorProductOnObjects( cat, b, DualOnObjects( cat, a ) ) );

end
],
Expand All @@ -650,27 +606,27 @@ AddFinalDerivationBundle( "deriving the internal hom by tensoring with the dual
[ DualOnObjects, 1 ] ],
function( cat, a, b )

return IdentityMorphism( cat, TensorProductOnObjects( cat, DualOnObjects( cat, a ), b ) );
return IdentityMorphism( cat, TensorProductOnObjects( cat, b, DualOnObjects( cat, a ) ) );

end
],
[
IsomorphismFromInternalHomIntoTensorUnitToDualObject,
[ [ RightUnitor, 1 ],
[ [ LeftUnitor, 1 ],
[ DualOnObjects, 1 ] ],
function( cat, object )

return RightUnitor( cat, DualOnObjects( cat, object ) );
return LeftUnitor( cat, DualOnObjects( cat, object ) );

end
],
[
IsomorphismFromDualObjectToInternalHomIntoTensorUnit,
[ [ RightUnitorInverse, 1 ],
[ [ LeftUnitorInverse, 1 ],
[ DualOnObjects, 1 ] ],
function( cat, object )

return RightUnitorInverse( cat, DualOnObjects( cat, object ) );
return LeftUnitorInverse( cat, DualOnObjects( cat, object ) );

end
] : CategoryFilter := IsRigidSymmetricClosedMonoidalCategory );
Loading
Loading