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7 Preordered sets viewed as categories
 7.1 Constructors
 7.2 Attributes
 7.3 GAP categories

7 Preordered sets viewed as categories

7.1 Constructors

7.1-1 ProsetAsCategory
‣ ProsetAsCategory( input_record )( operation )

Returns: a CAP category

Construct a thin category out of the input_record consisting of values to the keys:

7.1-2 DiscreteSkeletalCategory
‣ DiscreteSkeletalCategory( object_datum_filter_string )( operation )

Returns: a CAP category

Construct a discrete category out of

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 75 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 );
<(⊥)>

7.1-3 TotalOrderAsCategory
‣ TotalOrderAsCategory( object_datum_filter_string, object_datum_total_order_func )( operation )

Returns: a CAP category

Construct a totol order category out of

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 77 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

7.2 Attributes

7.2-1 UnderlyingObjectFilter
‣ UnderlyingObjectFilter( C )( attribute )

The object-membership function used to define the thin category C.

7.2-2 UnderlyingObjectMembershipFunction
‣ 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.

7.2-3 UnderlyingObjectEqualityFunction
‣ 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.

7.2-4 UnderlyingPreorderFunction
‣ 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.

7.2-5 UnderlyingDatum
‣ UnderlyingDatum( arg )( attribute )

7.3 GAP categories

7.3-1 IsProsetAsCategory
‣ IsProsetAsCategory( object )( filter )

Returns: true or false

The GAP category of preordered sets viewed as categories.

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