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1 Tools
 1.1 Tools
 1.2 Properties
 1.3 Attributes
 1.4 Functors
 1.5 Cell as evaluatable string

1 Tools

1.1 Tools

1.1-1 SET_RANGE_CATEGORY_Of_HOMOMORPHISM_STRUCTURE
‣ 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:

Furthermore, if IsCategoryWithDecidableLifts( H ) then

An error is raised if RangeCategoryOfHomomorphismStructure( C ) is already set.

1.1-2 ListKnownDoctrines
‣ ListKnownDoctrines( arg )( function )

Returns: a list

The function takes no arguments and returns the list of known doctrines.

1.1-3 ListMethodsOfDoctrine
‣ 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().

1.1-4 DummyCategoryInDoctrines
‣ 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

1.2 Properties

1.2-1 IsObjectFiniteCategory
‣ IsObjectFiniteCategory( C )( property )

Returns: true or false

The (evil) property of C being a category with finitely many objects.

1.2-2 IsFiniteCategory
‣ IsFiniteCategory( C )( property )

Returns: true or false

The (evil) property of C being a finite category.

1.2-3 IsEquivalentToFiniteCategory
‣ IsEquivalentToFiniteCategory( C )( property )

Returns: true or false

The property of C being equivalent to a finite category.

1.3 Attributes

1.3-1 SetOfObjectsOfCategory
‣ SetOfObjectsOfCategory( C )( attribute )

Returns: a list of CAP category objects

Return a duplicate free list of objects of the category C.

1.3-2 SetOfObjects
‣ 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.

1.3-3 SetOfMorphismsOfFiniteCategory
‣ SetOfMorphismsOfFiniteCategory( C )( attribute )

Returns: a list of a CAP category morphisms

Return a duplicate free list of morphisms of the finite category C.

1.3-4 SetOfMorphisms
‣ 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.

1.4 Functors

1.4-1 CovariantHomFunctor
‣ 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 ).

1.4-2 GlobalSectionFunctor
‣ 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 ).

1.5 Cell as evaluatable string

1.5-1 DatumOfCellAsEvaluatableString
‣ 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

The output string must, apart from the brackets, only consist of the substrings "ObjectConstructor", "MorphismConstructor" and the string in list_of_evaluatable_strings.

1.5-2 CellAsEvaluatableString
‣ 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

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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