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WithGivens for monoidal and (co)cartesian derivations #1494

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1b4122a
WithGiven* in derivations for Braiding
Oct 9, 2023
d8e4c21
WithGiven* in derivations of *Evaluation
Oct 3, 2023
a12bd07
WithGiven* in derivations of *Compose
Oct 9, 2023
9698af4
WithGiven* for monoidal Isomorphisms
Oct 6, 2023
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2 changes: 1 addition & 1 deletion CartesianCategories/PackageInfo.g
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Original file line number Diff line number Diff line change
Expand Up @@ -10,7 +10,7 @@ SetPackageInfo( rec(

PackageName := "CartesianCategories",
Subtitle := "Cartesian and cocartesian categories and various subdoctrines",
Version := "2023.11-02",
Version := "2023.11-03",
Date := ~.Version{[ 1 .. 10 ]},
Date := (function ( ) if IsBound( GAPInfo.SystemEnvironment.GAP_PKG_RELEASE_DATE ) then return GAPInfo.SystemEnvironment.GAP_PKG_RELEASE_DATE; else return Concatenation( ~.Version{[ 1 .. 4 ]}, "-", ~.Version{[ 6, 7 ]}, "-01" ); fi; end)( ),
License := "GPL-2.0-or-later",
Expand Down
13 changes: 7 additions & 6 deletions CartesianCategories/gap/BraidedCartesianCategoriesDerivedMethods.gi
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Expand Up @@ -11,22 +11,23 @@
AddDerivationToCAP( CartesianBraidingInverseWithGivenDirectProducts,
"CartesianBraidingInverseWithGivenDirectProducts as the inverse of the braiding",
[ [ InverseForMorphisms, 1 ],
[ CartesianBraiding, 1 ] ],
[ CartesianBraidingWithGivenDirectProducts, 1 ] ],

function( cat, object_2_x_object_1, object_1, object_2, object_1_x_object_2 )
##TODO: Use CartesianBraidingWithGiven
return InverseForMorphisms( cat, CartesianBraiding( cat, object_1, object_2 ) );

return InverseForMorphisms( cat, CartesianBraidingWithGivenDirectProducts( cat, object_1_x_object_2, object_1, object_2, object_2_x_object_1 ) );

end : CategoryFilter := IsCartesianCategory );

##
AddDerivationToCAP( CartesianBraidingWithGivenDirectProducts,
"CartesianBraidingWithGivenDirectProducts as the inverse of CartesianBraidingInverse",
[ [ InverseForMorphisms, 1 ],
[ CartesianBraidingInverse, 1 ] ],
[ CartesianBraidingInverseWithGivenDirectProducts, 1 ] ],

function( cat, object_1_x_object_2, object_1, object_2, object_2_x_object_1 )
##TODO: Use CartesianBraidingInverseWithGiven
return InverseForMorphisms( cat, CartesianBraidingInverse( cat, object_1, object_2 ) );

return InverseForMorphisms( cat, CartesianBraidingInverseWithGivenDirectProducts( cat, object_2_x_object_1, object_1, object_2, object_1_x_object_2 ) );

end : CategoryFilter := IsCartesianCategory );

13 changes: 7 additions & 6 deletions CartesianCategories/gap/BraidedCocartesianCategoriesDerivedMethods.gi
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Expand Up @@ -11,22 +11,23 @@
AddDerivationToCAP( CocartesianBraidingInverseWithGivenCoproducts,
"CocartesianBraidingInverseWithGivenCoproducts as the inverse of the braiding",
[ [ InverseForMorphisms, 1 ],
[ CocartesianBraiding, 1 ] ],
[ CocartesianBraidingWithGivenCoproducts, 1 ] ],

function( cat, object_2_u_object_1, object_1, object_2, object_1_u_object_2 )
##TODO: Use CocartesianBraidingWithGiven
return InverseForMorphisms( cat, CocartesianBraiding( cat, object_1, object_2 ) );

return InverseForMorphisms( cat, CocartesianBraidingWithGivenCoproducts( cat, object_1_u_object_2, object_1, object_2, object_2_u_object_1 ) );

end : CategoryFilter := IsCocartesianCategory );

##
AddDerivationToCAP( CocartesianBraidingWithGivenCoproducts,
"CocartesianBraidingWithGivenCoproducts as the inverse of CocartesianBraidingInverse",
[ [ InverseForMorphisms, 1 ],
[ CocartesianBraidingInverse, 1 ] ],
[ CocartesianBraidingInverseWithGivenCoproducts, 1 ] ],

function( cat, object_1_u_object_2, object_1, object_2, object_2_u_object_1 )
##TODO: Use CocartesianBraidingInverseWithGiven
return InverseForMorphisms( cat, CocartesianBraidingInverse( cat, object_1, object_2 ) );

return InverseForMorphisms( cat, CocartesianBraidingInverseWithGivenCoproducts( cat, object_2_u_object_1, object_1, object_2, object_1_u_object_2 ) );

end : CategoryFilter := IsCocartesianCategory );

103 changes: 67 additions & 36 deletions CartesianCategories/gap/SymmetricCartesianClosedCategoriesDerivedMethods.gi
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Expand Up @@ -291,7 +291,7 @@ end : CategoryFilter := IsCartesianClosedCategory );
##
AddDerivationToCAP( CartesianEvaluationForCartesianDualWithGivenDirectProduct,
"CartesianEvaluationForCartesianDualWithGivenDirectProduct using the direct product-exponential adjunction and IsomorphismFromCartesianDualObjectToExponentialIntoTerminalObject",
[ [ ExponentialToDirectProductAdjunctionMap, 1 ],
[ [ ExponentialToDirectProductAdjunctionMapWithGivenDirectProduct, 1 ],
[ IsomorphismFromCartesianDualObjectToExponentialIntoTerminalObject, 1 ] ],

function( cat, s, a, r )
Expand All @@ -300,8 +300,11 @@ AddDerivationToCAP( CartesianEvaluationForCartesianDualWithGivenDirectProduct,
#
# Adjoint( a^v → Exp(a,1) ) = ( a^v × a → 1 )

return ExponentialToDirectProductAdjunctionMap( cat, a, r,
IsomorphismFromCartesianDualObjectToExponentialIntoTerminalObject( cat, a ) );
return ExponentialToDirectProductAdjunctionMapWithGivenDirectProduct( cat,
a,
r,
IsomorphismFromCartesianDualObjectToExponentialIntoTerminalObject( cat, a ),
s );

end : CategoryFilter := IsCartesianClosedCategory );

Expand Down Expand Up @@ -531,14 +534,16 @@ end : CategoryFilter := IsCartesianClosedCategory );
AddDerivationToCAP( IsomorphismFromObjectToExponentialWithGivenExponential,
"IsomorphismFromObjectToExponentialWithGivenExponential using the coevaluation morphism",
[ [ TerminalObject, 1 ],
[ DirectProduct, 1 ],
[ ExponentialOnObjects, 1 ],
[ PreCompose, 1 ],
[ CartesianCoevaluationMorphism, 1 ],
[ ExponentialOnMorphisms, 1 ],
[ CartesianCoevaluationMorphismWithGivenRange, 1 ],
[ ExponentialOnMorphismsWithGivenExponentials, 1 ],
[ IdentityMorphism, 1 ],
[ CartesianRightUnitor, 1 ] ],
[ CartesianRightUnitorWithGivenDirectProduct, 1 ] ],

function( cat, a, internal_hom )
local unit;
local unit, a_x_1, exp_1_ax1;

# a
# |
Expand All @@ -551,20 +556,24 @@ AddDerivationToCAP( IsomorphismFromObjectToExponentialWithGivenExponential,
# Exp(1, a)

unit := TerminalObject( cat );
a_x_1 := BinaryDirectProduct( cat, a, unit );
exp_1_ax1:= ExponentialOnObjects( cat, unit, a_x_1 );

return PreCompose( cat,
CartesianCoevaluationMorphism( cat, a, unit ),
ExponentialOnMorphisms( cat,
IdentityMorphism( cat, unit ),
CartesianRightUnitor( cat, a ) ) );
CartesianCoevaluationMorphismWithGivenRange( cat, a, unit, exp_1_ax1 ),
ExponentialOnMorphismsWithGivenExponentials( cat,
exp_1_ax1,
IdentityMorphism( cat, unit ),
CartesianRightUnitorWithGivenDirectProduct( cat, a, a_x_1 ),
internal_hom ) );

end : CategoryFilter := IsCartesianClosedCategory );

##
AddDerivationToCAP( IsomorphismFromObjectToExponentialWithGivenExponential,
"IsomorphismFromObjectToExponentialWithGivenExponential as the adjoint of the right unitor",
[ [ TerminalObject, 1 ],
[ DirectProductToExponentialAdjunctionMap, 1 ],
[ DirectProductToExponentialAdjunctionMapWithGivenExponential, 1 ],
[ CartesianRightUnitor, 1 ] ],

function( cat, a, internal_hom )
Expand All @@ -573,10 +582,11 @@ AddDerivationToCAP( IsomorphismFromObjectToExponentialWithGivenExponential,
#
# Adjoint( ρ_a ) = ( a → Exp(1,a) )

return DirectProductToExponentialAdjunctionMap( cat,
return DirectProductToExponentialAdjunctionMapWithGivenExponential( cat,
a,
TerminalObject( cat ),
CartesianRightUnitor( cat, a ) );
CartesianRightUnitor( cat, a ),
internal_hom );

end : CategoryFilter := IsCartesianClosedCategory );

Expand All @@ -598,10 +608,13 @@ AddDerivationToCAP( IsomorphismFromExponentialToObjectWithGivenExponential,
"IsomorphismFromExponentialToObjectWithGivenExponential using the evaluation morphism",
[ [ TerminalObject, 1 ],
[ PreCompose, 1 ],
[ CartesianRightUnitorInverse, 1 ],
[ CartesianEvaluationMorphism, 1 ] ],
[ DirectProduct, 1 ],
[ ExponentialOnObjects, 1 ],
[ CartesianRightUnitorInverseWithGivenDirectProduct, 1 ],
[ CartesianEvaluationMorphismWithGivenSource, 1 ] ],

function( cat, a, internal_hom )
local unit, exp_1a, exp_1a_x_1;

# Exp(1,a)
# |
Expand All @@ -613,9 +626,13 @@ AddDerivationToCAP( IsomorphismFromExponentialToObjectWithGivenExponential,
# v
# a

unit := TerminalObject( cat );
exp_1a := ExponentialOnObjects( cat, unit, a );
exp_1a_x_1 := BinaryDirectProduct( cat, exp_1a, unit );

return PreCompose( cat,
CartesianRightUnitorInverse( cat, internal_hom ),
CartesianEvaluationMorphism( cat, TerminalObject( cat ), a ) );
CartesianRightUnitorInverseWithGivenDirectProduct( cat, internal_hom, exp_1a_x_1 ),
CartesianEvaluationMorphismWithGivenSource( cat, unit, a, exp_1a_x_1 ) );

end : CategoryFilter := IsCartesianClosedCategory );

Expand Down Expand Up @@ -682,34 +699,38 @@ end : CategoryFilter := IsCartesianClosedCategory );
##
AddDerivationToCAP( CartesianEvaluationMorphismWithGivenSource,
"CartesianEvaluationMorphismWithGivenSource using the direct product-exponential adjunction on the identity",
[ [ ExponentialToDirectProductAdjunctionMap, 1 ],
[ [ ExponentialToDirectProductAdjunctionMapWithGivenDirectProduct, 1 ],
[ IdentityMorphism, 1 ],
[ ExponentialOnObjects, 1 ] ],

function( cat, a, b, direct_product_object )

# Adjoint( id_Exp(a,b): Exp(a,b) → Exp(a,b) ) = ( Exp(a,b) × a → b )

return ExponentialToDirectProductAdjunctionMap( cat,
a, b,
IdentityMorphism( cat, ExponentialOnObjects( cat, a, b ) ) );
return ExponentialToDirectProductAdjunctionMapWithGivenDirectProduct( cat,
a,
b,
IdentityMorphism( cat, ExponentialOnObjects( cat, a, b ) ),
direct_product_object );

end : CategoryFilter := IsCartesianClosedCategory );

##
AddDerivationToCAP( CartesianCoevaluationMorphismWithGivenRange,
"CartesianCoevaluationMorphismWithGivenRange using the direct product-exponential adjunction on the identity",
[ [ DirectProductToExponentialAdjunctionMap, 1 ],
[ [ DirectProductToExponentialAdjunctionMapWithGivenExponential, 1 ],
[ IdentityMorphism, 1 ],
[ DirectProduct, 1 ] ],

function( cat, a, b, internal_hom )

# Adjoint( id_(a × b): a × b → a × b ) = ( a → Exp(b, a × b) )

return DirectProductToExponentialAdjunctionMap( cat,
a, b,
IdentityMorphism( cat, BinaryDirectProduct( cat, a, b ) ) );
return DirectProductToExponentialAdjunctionMapWithGivenExponential( cat,
a,
b,
IdentityMorphism( cat, BinaryDirectProduct( cat, a, b ) ),
internal_hom );

end : CategoryFilter := IsCartesianClosedCategory );

Expand Down Expand Up @@ -852,13 +873,14 @@ end : CategoryFilter := IsCartesianClosedCategory );
##
AddDerivationToCAP( CartesianPostComposeMorphismWithGivenObjects,
"CartesianPostComposeMorphismWithGivenObjects using CartesianPreComposeMorphism and braiding",
[ [ CartesianBraiding, 1 ],
[ [ CartesianBraidingWithGivenDirectProducts, 1 ],
[ ExponentialOnObjects, 2 ],
[ PreCompose, 1 ],
[ CartesianPreComposeMorphism, 1 ] ],
[ DirectProduct, 1],
[ CartesianPreComposeMorphismWithGivenObjects, 1 ] ],

function( cat, source, a, b, c, range )
local braiding;
local exp_ab, exp_bc, exp_ab_x_exp_bc, braiding;

# Exp(b,c) × Exp(a,b)
# |
Expand All @@ -870,22 +892,27 @@ AddDerivationToCAP( CartesianPostComposeMorphismWithGivenObjects,
# v
# Exp(a,c)

braiding := CartesianBraiding( cat, ExponentialOnObjects( cat, b, c ), ExponentialOnObjects( cat, a, b ) );
exp_ab := ExponentialOnObjects( cat, a, b );
exp_bc := ExponentialOnObjects( cat, b, c );
exp_ab_x_exp_bc := BinaryDirectProduct( cat, exp_ab, exp_bc );

braiding := CartesianBraidingWithGivenDirectProducts( cat, source, exp_bc, exp_ab, exp_ab_x_exp_bc );

return PreCompose( cat, braiding, CartesianPreComposeMorphism( cat, a, b, c ) );
return PreCompose( cat, braiding, CartesianPreComposeMorphismWithGivenObjects( cat, exp_ab_x_exp_bc, a, b, c, range ) );

end : CategoryFilter := IsCartesianClosedCategory );

##
AddDerivationToCAP( CartesianPreComposeMorphismWithGivenObjects,
"CartesianPreComposeMorphismWithGivenObjects using CartesianPostComposeMorphism and braiding",
[ [ CartesianBraiding, 1 ],
[ [ CartesianBraidingWithGivenDirectProducts, 1 ],
[ ExponentialOnObjects, 2 ],
[ PreCompose, 1 ],
[ CartesianPostComposeMorphism, 1 ] ],
[ DirectProduct, 1],
[ CartesianPostComposeMorphismWithGivenObjects, 1 ] ],

function( cat, source, a, b, c, range )
local braiding;
local exp_ab, exp_bc, exp_bc_x_exp_ab, braiding;

# Exp(a,b) × Exp(b,c)
# |
Expand All @@ -897,9 +924,13 @@ AddDerivationToCAP( CartesianPreComposeMorphismWithGivenObjects,
# v
# Exp(a,c)

braiding := CartesianBraiding( cat, ExponentialOnObjects( cat, a, b ), ExponentialOnObjects( cat, b, c ) );
exp_ab := ExponentialOnObjects( cat, a, b );
exp_bc := ExponentialOnObjects( cat, b, c );
exp_bc_x_exp_ab := BinaryDirectProduct( cat, exp_bc, exp_ab );

braiding := CartesianBraidingWithGivenDirectProducts( cat, source, exp_bc, exp_ab, exp_bc_x_exp_ab );

return PreCompose( cat, braiding, CartesianPostComposeMorphism( cat, a, b, c ) );
return PreCompose( cat, braiding, CartesianPostComposeMorphismWithGivenObjects( cat, exp_bc_x_exp_ab, a, b, c, range ) );

end : CategoryFilter := IsCartesianClosedCategory );

Expand Down
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