# Realizations Covariant Functorial Construction¶

class sage.categories.realizations.Category_realization_of_parent(parent_with_realization)

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()


Todo

Add an optional argument to allow for:

sage: Realizations(A, category = Blahs()) # todo: not implemented

class sage.categories.realizations.RealizationsCategory(category, *args)

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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