‣ ProsetAsCategory( input_record ) | ( operation ) |
Returns: a CAP category
Construct a thin category out of the input_record consisting of values to the keys:
name: the string defining the name of the category,
object_datum_filter_string: a string such that ValueGlobal( object_datum_filter_string ) is the GAP-filter of the underlying objects,
object_datum_membership_func: the underlying-object-membership function,
object_datum_equality_func: the underlying-object-equality function,
object_datum_preorder_func: the preorder function on the underlying objects, and
properties: a list of categorical properties, additional to IsThinCategory.
If name is missing it defaults to object_datum_filter_string.
If object_datum_membership_func is missing it defaults to ValueGlobal( object_datum_filter_string ).
If object_datum_equality_func is missing it defaults to EQ.
If object_datum_preorder_func is missing it defaults to object_datum_equality_func.
If properties is missing it defaults to the empty list.
‣ DiscreteSkeletalCategory( object_datum_filter_string ) | ( operation ) |
Returns: a CAP category
Construct a discrete category out of
the GAP-filter ValueGlobal( object_datum_filter_string ) of the underlying objects.
gap> LoadPackage( "Locales" ); true gap> D := DiscreteSkeletalCategory( "IsPosInt" ); DiscreteSkeletalCategory( "IsPosInt" ) gap> Display( D ); A CAP category with name DiscreteSkeletalCategory( "IsPosInt" ): 8 primitive operations were used to derive 77 operations for this category which algorithmically * IsCategoryWithDecidableColifts * IsCategoryWithDecidableLifts * IsPosetCategory * IsEquippedWithHomomorphismStructure and furthermore mathematically * IsDiscreteCategory gap> one := 1 / D; <An object in DiscreteSkeletalCategory( "IsPosInt" )> gap> Display( one ); 1 An object in DiscreteSkeletalCategory( "IsPosInt" ) given by the above data gap> IsWellDefined( one ); true gap> two := 2 / D; <An object in DiscreteSkeletalCategory( "IsPosInt" )> gap> id_one := IdentityMorphism( one ); <An identity morphism in DiscreteSkeletalCategory( "IsPosInt" )> gap> MorphismDatum( id_one ); fail gap> IsOne( PreCompose( id_one, id_one ) ); true gap> IsOne( UniqueMorphism( one, one ) ); true gap> IsWellDefined( UniqueMorphism( one, two ) ); false gap> HomStructure( one, one ); <(⊤)> gap> HomStructure( one, two ); <(⊥)> gap> HomStructure( two, one ); <(⊥)>
‣ TotalOrderAsCategory( object_datum_filter_string, object_datum_total_order_func ) | ( operation ) |
Returns: a CAP category
Construct a totol order category out of
the GAP-filter ValueGlobal( object_datum_filter_string ) of the underlying objects, and
the total order function on the underlying objects.
gap> LoadPackage( "Locales" ); true gap> D := TotalOrderAsCategory( "IsInt", {a,b} -> a <= b ); TotalOrderAsCategory( "IsInt" ) gap> Display( D ); A CAP category with name TotalOrderAsCategory( "IsInt" ): 8 primitive operations were used to derive 79 operations for this category which algorithmically * IsCategoryWithDecidableColifts * IsCategoryWithDecidableLifts * IsPosetCategory * IsEquippedWithHomomorphismStructure and furthermore mathematically * IsTotalOrderCategory gap> one := 1 / D; <An object in TotalOrderAsCategory( "IsInt" )> gap> Display( one ); 1 An object in TotalOrderAsCategory( "IsInt" ) given by the above data gap> IsWellDefined( one ); true gap> IsWellDefined( -1 / D ); true gap> two := 2 / D; <An object in TotalOrderAsCategory( "IsInt" )> gap> id_one := IdentityMorphism( one ); <An identity morphism in TotalOrderAsCategory( "IsInt" )> gap> MorphismDatum( id_one ); fail gap> IsOne( PreCompose( id_one, id_one ) ); true gap> IsOne( UniqueMorphism( one, one ) ); true gap> IsWellDefined( UniqueMorphism( one, two ) ); true gap> HomStructure( one, one ); <(⊤)> gap> IsHomSetInhabited( one, one ); true gap> HomStructure( one, two ); <(⊤)> gap> IsHomSetInhabited( one, two ); true gap> HomStructure( two, one ); <(⊥)> gap> IsHomSetInhabited( two, one ); false gap> DirectProduct( [ one, two ] ) = one; true gap> Coproduct( [ one, two ] ) = two; true
‣ UnderlyingObjectFilter( C ) | ( attribute ) |
The object-membership function used to define the thin category C.
‣ UnderlyingObjectMembershipFunction( C ) | ( attribute ) |
The underlying-object-membership function used to define the thin category C. This function accepts two arguments in the GAP-filter ValueGlobal(UnderlyingObjectFilter(C)) and returns true if the underlying objects are equal false.
‣ UnderlyingObjectEqualityFunction( C ) | ( attribute ) |
The underlying-object-equality function used to define the thin category C. This function accepts two arguments in the GAP-filter ValueGlobal(UnderlyingObjectFilter(C)) and returns true if the underlying objects are equal false.
‣ UnderlyingPreorderFunction( C ) | ( attribute ) |
The preorder function used to define the thin category C. This function accepts two arguments in the GAP-filter ValueGlobal(UnderlyingObjectFilter(C)) and returns true or false.
‣ UnderlyingObject( arg ) | ( attribute ) |
‣ IsProsetAsCategory( object ) | ( filter ) |
Returns: true or false
The GAP category of preordered sets viewed as categories.
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