‣ SET_RANGE_CATEGORY_Of_HOMOMORPHISM_STRUCTURE( C, H ) | ( operation ) |
The input are two categories C and H. There is not output but the following side effects are applied to C:
SetRangeCategoryOfHomomorphismStructure( C, H )
SetIsEquippedWithHomomorphismStructure( C, true )
Furthermore, if IsCategoryWithDecidableLifts( H ) then
SetIsCategoryWithDecidableLifts( C, true )
SetIsCategoryWithDecidableColifts( C, true )
An error is raised if RangeCategoryOfHomomorphismStructure( C ) is already set.
‣ ListKnownDoctrines( arg ) | ( function ) |
Returns: a list
The function takes no arguments and returns the list of known doctrines.
gap> LoadPackage( "FunctorCategories", false ); true gap> ListKnownDoctrines( ); [ "IsCapCategory", \ "IsEnrichedOverCommutativeRegularSemigroup", \ "IsAbCategory", \ "IsCategoryWithTerminalObject", \ "IsCartesianCategory", \ "IsCategoryWithInitialObject", "IsCocartesianCategory", \ "IsBicartesianCategory", \ "IsCategoryWithZeroObject", \ "IsAdditiveCategory", \ "IsCategoryWithCoequalizers", \ "IsCategoryWithCokernels", \ "IsCategoryWithEqualizers", "IsCategoryWithKernels", \ "IsFiniteCocompleteCategory", \ "IsFiniteCompleteCategory", \ "IsFiniteBicompleteCategory", \ "IsPreAbelianCategory", \ "IsAbelianCategory", "IsAbelianCategoryWithEnoughInjectives", \ "IsAbelianCategoryWithEnoughProjectives", \ "IsCartesianClosedCategory", \ "IsDistributiveCategory", "IsBicartesianClosedCategory", \ "IsCocartesianCoclosedCategory", \ "IsCodistributiveCategory", \ "IsBicartesianCoclosedCategory", \ "IsThinCategory", \ "IsCartesianProset", "IsCocartesianProset", \ "IsBicartesianProset", \ "IsDistributiveBicartesianProset", \ "IsCoHeytingAlgebroid", \ "IsHeytingAlgebroid", \ "IsBiHeytingAlgebroid", "IsPosetCategory", \ "IsJoinSemiLattice", \ "IsMeetSemiLattice", \ "IsLattice", \ "IsDistributiveLattice", \ "IsCoHeytingAlgebra", \ "IsHeytingAlgebra", \ "IsBiHeytingAlgebra", "IsBooleanAlgebroid", \ "IsBooleanAlgebra", \ "IsMonoidalCategory", \ "IsSymmetricMonoidalCategory", \ "IsCategoryWithDecidableColifts", \ "IsCategoryWithDecidableLifts", "IsClosedMonoidalCategory", \ "IsMonoidalProset", \ "IsClosedMonoidalProset", \ "IsMonoidalPoset", \ "IsClosedMonoidalPoset", \ "IsMonoidalLattice", \ "IsClosedMonoidalLattice", "IsCoclosedMonoidalCategory", \ "IsCoclosedMonoidalProset", \ "IsCoclosedMonoidalPoset", \ "IsCoclosedMonoidalLattice", \ "IsElementaryTopos", "IsEquippedWithHomomorphismStructure", \ "IsObjectFiniteCategory", \ "IsFinitelyPresentedCategory", \ "IsFiniteCategory", \ "IsLeftClosedMonoidalCategory", "IsLeftCoclosedMonoidalCategory", \ "IsLinearCategoryOverCommutativeRing", \ "IsLinearCategoryOverCommutativeRingWithFinitelyGeneratedFreeExternalHoms", "IsSymmetricClosedMonoidalCategory", \ "IsRigidSymmetricClosedMonoidalCategory", \ "IsSymmetricCoclosedMonoidalCategory", \ "IsRigidSymmetricCoclosedMonoidalCategory", "IsSymmetricMonoidalProset", \ "IsSymmetricClosedMonoidalProset", \ "IsSymmetricMonoidalPoset", \ "IsSymmetricClosedMonoidalPoset", \ "IsSymmetricMonoidalLattice", "IsSymmetricClosedMonoidalLattice", \ "IsSymmetricCoclosedMonoidalProset", \ "IsSymmetricCoclosedMonoidalPoset", \ "IsSymmetricCoclosedMonoidalLattice" ]
‣ DummyCategoryInDoctrines( doctrine_names ) | ( operation ) |
Returns: a CAP category
The argument is a nonempty list doctrine_names of strings of doctrine names and the output is a dummy category created using DummyCategory which lies in the given doctrine. If the option minimal is set to true, then only a logically minimal subset of the doctrines is considered.
gap> LoadPackage( "ToolsForCategoricalTowers", false ); true gap> D1 := DummyCategoryInDoctrines( "IsAbCategory" ); DummyCategoryInDoctrines( [ "IsAbCategory" ] ) gap> Display( D1 ); A CAP category with name DummyCategoryInDoctrines( [ "IsAbCategory" ] ): 16 primitive operations were used to derive 28 operations for this category which algorithmically * IsAbCategory gap> D2 := DummyCategoryInDoctrines( [ "IsAbCategory", "IsAbelianCategory" ] ); DummyCategoryInDoctrines( [ "IsAbelianCategory" ] ) gap> Display( D2 ); A CAP category with name DummyCategoryInDoctrines( [ "IsAbelianCategory" ] ): 33 primitive operations were used to derive 291 operations for this category which algorithmically * IsAbelianCategory gap> D3 := DummyCategoryInDoctrines( > [ "IsCategoryWithInitialObject", > "IsCategoryWithTerminalObject", > "IsCategoryWithZeroObject" ] ); DummyCategoryInDoctrines( [ "IsCategoryWithInitialObject", "IsCategoryWithTerminalObject", "IsCategoryWithZeroObject" ] ) gap> Display( D3 ); A CAP category with name DummyCategoryInDoctrines( [ "IsCategoryWithInitialObject", "IsCategoryWithTerminalObject", "IsCategoryWithZeroObject" ] ): 18 primitive operations were used to derive 41 operations for this category which algorithmically * IsCategoryWithZeroObject gap> D4 := DummyCategoryInDoctrines( > [ "IsCategoryWithInitialObject", > "IsCategoryWithTerminalObject", > "IsCategoryWithZeroObject" ] : minimal := true ); DummyCategoryInDoctrines( [ "IsCategoryWithZeroObject" ] ) gap> Display( D4 ); A CAP category with name DummyCategoryInDoctrines( [ "IsCategoryWithZeroObject" ] ): 14 primitive operations were used to derive 41 operations for this category which algorithmically * IsCategoryWithZeroObject
‣ IsObjectFiniteCategory( C ) | ( property ) |
Returns: true or false
The (evil) property of C being a category with finitely many objects.
‣ IsFiniteCategory( C ) | ( property ) |
Returns: true or false
The (evil) property of C being a finite category.
‣ IsEquivalentToFiniteCategory( C ) | ( property ) |
Returns: true or false
The property of C being equivalent to a finite category.
‣ SetOfObjectsOfCategory( C ) | ( attribute ) |
Returns: a list of CAP category objects
Return a duplicate free list of objects of the category C.
‣ SetOfObjects( C ) | ( attribute ) |
Returns: a list of CAP category objects
Return a duplicate free list of objects of the category C. The corresponding CAP operation is SetOfObjectsOfCategory.
‣ SetOfMorphismsOfFiniteCategory( C ) | ( attribute ) |
Returns: a list of a CAP category morphisms
Return a duplicate free list of morphisms of the finite category C.
‣ SetOfMorphisms( C ) | ( attribute ) |
Returns: a list of CAP category objects
Return a duplicate free list of morphisms of the finite category C. The corresponding CAP operation is SetOfMorphismsOfFiniteCategory.
‣ CovariantHomFunctor( o ) | ( attribute ) |
Returns: a CAP functor
The input is an object o in a category \(C\). The output is the covariant Hom functor \(\mathrm{Hom}\)(o,-) from the category \(C\) to RangeCategoryOfHomomorphismStructure( C ).
‣ GlobalSectionFunctor( C ) | ( attribute ) |
Returns: a CAP functor
Returns the global section functor \(\mathrm{Hom}(1,-)\) from the category C with terminal object \(1\) to RangeCategoryOfHomomorphismStructure( C ).
‣ DatumOfCellAsEvaluatableString( c, list_of_evaluatable_strings ) | ( operation ) |
Returns: a string
The arguments is a category cell c and a list list_of_evaluatable_strings of string all which must be evalutable with EvalString. The output is a string str such that
EvalString( str ) = ObjectDatum( c ) if c is a category object, or
EvalString( str ) = MorphismDatum( c ) if c is a category morphism.
The output string must, apart from the brackets, only consist of the substrings "ObjectConstructor", "MorphismConstructor" and the string in list_of_evaluatable_strings.
‣ CellAsEvaluatableString( c, list_of_evaluatable_strings ) | ( operation ) |
Returns: a string
The arguments is a category cell c and a list list_of_evaluatable_strings of string all which must be evalutable with EvalString. The output is a string str such that
IsEqualForObjects( EvalString( str ), c ) if c is a category object, or
IsEqualForMorphismsOnMor( EvalString( str ), c ) if c is a category morphism.
The output string must, apart from the brackets, only consist of the substrings "ObjectConstructor", "MorphismConstructor" and the string in list_of_evaluatable_strings.
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