‣ 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.
‣ ListMethodsOfDoctrine( doctrine_name ) | ( function ) |
Returns: a list
The argument doctrine_name is a string which is a name of a valid doctrine in the ListKnownDoctrines().
‣ 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" ] ): 32 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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