Realizations Covariant Functorial Construction

class sage.categories.realizations.Category_realization_of_parent(parent_with_realization)[source]¶

Bases: Category_over_base, BindableClass

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

INPUT:

parent_with_realization – a parent

See also

Sets().WithRealizations

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> A = Sets().WithRealizations().example(); A                                # needs sage.modules
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

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

Python

>>> from sage.all import *
>>> C = A.Realizations(); C                                                   # needs sage.modules
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)[source]¶

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

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

Python

>>> from sage.all import *
>>> Algebras(QQ).Realizations()
Join of Category of algebras over Rational Field
    and Category of realizations of unital magmas

The category of realizations of a given algebra:

Sage

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

Python

>>> from sage.all import *
>>> A = Sets().WithRealizations().example(); A                                # needs sage.modules
The subset algebra of {1, 2, 3} over Rational Field
>>> A.Realizations()                                                          # needs sage.modules
Category of realizations of
 The subset algebra of {1, 2, 3} over Rational Field

>>> 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
>>> C.super_categories()
[Category of graded Hopf algebras with basis over Rational Field,
 Category of realizations of Hopf algebras over Rational Field]

>>> TestSuite(C).run()

Realizations Covariant Functorial Construction¶