Realizations Covariant Functorial Construction

See also

class sage.categories.realizations.Category_realization_of_parent(parent_with_realization)

Bases: sage.categories.category_types.Category_over_base, sage.misc.bindable_class.BindableClass

An abstract base class for categories of all realizations of a given parent

INPUT:

  • parent_with_realization – a parent

EXAMPLES:

sage: A = Sets().WithRealizations().example(); A
The subset algebra of {1, 2, 3} over Rational Field

The role of this base class is to implement some technical goodies, like the binding A.Realizations() when a subclass Realizations is implemented as a nested class in A (see the code of the example):

sage: C = A.Realizations(); C
Category of realizations of The subset algebra of {1, 2, 3} over Rational Field

as well as the name for that category.

sage.categories.realizations.Realizations(self)

Return the category of realizations of the parent self or of objects of the category self

INPUT:

  • self – a parent or a concrete category

Note

this function is actually inserted as a method in the class Category (see Realizations()). It is defined here for code locality reasons.

EXAMPLES:

The category of realizations of some algebra:

sage: Algebras(QQ).Realizations()
Join of Category of algebras over Rational Field and Category of realizations of magmas

The category of realizations of a given algebra:

sage: A = Sets().WithRealizations().example(); A
The subset algebra of {1, 2, 3} over Rational Field
sage: A.Realizations()
Category of realizations of The subset algebra of {1, 2, 3} over Rational Field

sage: C = GradedHopfAlgebrasWithBasis(QQ).Realizations(); C
Join of Category of graded hopf algebras with basis over Rational Field and Category of realizations of hopf algebras over Rational Field
sage: C.super_categories()
[Category of graded hopf algebras with basis over Rational Field, Category of realizations of hopf algebras over Rational Field]

sage: TestSuite(C).run()

See also

Todo

Add an optional argument to allow for:

sage: Realizations(A, category = Blahs()) # todo: not implemented
class sage.categories.realizations.RealizationsCategory(category, *args)

Bases: sage.categories.covariant_functorial_construction.RegressiveCovariantConstructionCategory

An abstract base class for all categories of realizations category

Relization are implemented as RegressiveCovariantConstructionCategory. See there for the documentation of how the various bindings such as Sets().Realizations() and P.Realizations(), where P is a parent, work.

TESTS:

sage: Sets().Realizations
<bound method Sets_with_category.Realizations of Category of sets>
sage: Sets().Realizations()
Category of realizations of sets
sage: Sets().Realizations().super_categories()
[Category of sets]
sage: Groups().Realizations().super_categories()
[Category of groups, Category of realizations of magmas]

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