With Realizations Covariant Functorial Construction

See also

sage.categories.with_realizations.WithRealizations(self)

Returns the category of parents in self endowed with multiple realizations

INPUT:

  • self – a category

See also

Note

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

EXAMPLES:

sage: Sets().WithRealizations()
Category of sets with realizations

Parent with realizations

Let us now explain the concept of realizations. A parent with realizations is a facade parent (see Sets.Facades) admitting multiple concrete realizations where its elements are represented. Consider for example an algebra \(A\) which admits several natural bases:

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

For each such basis \(B\) one implements a parent \(P_B\) which realizes \(A\) with its elements represented by expanding them on the basis \(B\):

sage: A.F()
The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis
sage: A.Out()
The subset algebra of {1, 2, 3} over Rational Field in the Out basis
sage: A.In()
The subset algebra of {1, 2, 3} over Rational Field in the In basis

sage: A.an_element()
F[{}] + 2*F[{1}] + 3*F[{2}] + F[{1, 2}]

If \(B\) and \(B'\) are two bases, then the change of basis from \(B\) to \(B'\) is implemented by a canonical coercion between \(P_B\) and \(P_{B'}\):

sage: F = A.F(); In = A.In(); Out = A.Out()
sage: i = In.an_element(); i
In[{}] + 2*In[{1}] + 3*In[{2}] + In[{1, 2}]
sage: F(i)
7*F[{}] + 3*F[{1}] + 4*F[{2}] + F[{1, 2}]
sage: F.coerce_map_from(Out)
Generic morphism:
  From: The subset algebra of {1, 2, 3} over Rational Field in the Out basis
  To:   The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis

allowing for mixed arithmetic:

sage: (1 + Out.from_set(1)) * In.from_set(2,3)
Out[{}] + 2*Out[{1}] + 2*Out[{2}] + 2*Out[{3}] + 2*Out[{1, 2}] + 2*Out[{1, 3}] + 4*Out[{2, 3}] + 4*Out[{1, 2, 3}]

In our example, there are three realizations:

sage: A.realizations()
[The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis,
 The subset algebra of {1, 2, 3} over Rational Field in the In basis,
 The subset algebra of {1, 2, 3} over Rational Field in the Out basis]

The set of all realizations of \(A\), together with the coercion morphisms is a category (whose class inherits from Category_realization_of_parent):

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

The various parent realizing \(A\) belong to this category:

sage: A.F() in A.Realizations()
True

\(A\) itself is in the category of algebras with realizations:

sage: A in Algebras(QQ).WithRealizations()
True

The (mostly technical) WithRealizations categories are the analogs of the *WithSeveralBases categories in MuPAD-Combinat. They provide support tools for handling the different realizations and the morphisms between them.

Typically, FiniteDimensionalVectorSpaces(QQ).WithRealizations() will eventually be in charge, whenever a coercion \(\phi: A\mapsto B\) is registered, to register \(\phi^{-1}\) as coercion \(B \mapsto A\) if there is none defined yet. To achieve this, FiniteDimensionalVectorSpaces would provide a nested class WithRealizations implementing the appropriate logic.

WithRealizations is a regressive covariant functorial construction. On our example, this simply means that \(A\) is automatically in the category of rings with realizations (covariance):

sage: A in Rings().WithRealizations()
True

and in the category of algebras (regressiveness):

sage: A in Algebras(QQ)
True

Note

For C a category, C.WithRealizations() in fact calls sage.categories.with_realizations.Realizations(C). The later is responsible for building the hierarchy of the categories with realizations in parallel to that of their base categories, optimizing away those categories that do not provide a WithRealizations nested class. See sage.categories.covariant_functorial_construction for the technical details.

Note

Design question: currently WithRealizations is a regressive construction. That is self.WithRealizations() is a subcategory of self by default:

sage: Algebras(QQ).WithRealizations().super_categories()
[Category of algebras over Rational Field, Category of monoids with realizations, Category of commutative additive monoids with realizations]

Is this always desirable? For example, AlgebrasWithBasis(QQ).WithRealizations() should certainly be a subcategory of Algebras(QQ), but not of AlgebrasWithBasis(QQ). This is because AlgebrasWithBasis(QQ) is specifying something about the concrete realization.

TESTS:

sage: Semigroups().WithRealizations()
Join of Category of semigroups and Category of sets with realizations
sage: C = GradedHopfAlgebrasWithBasis(QQ).WithRealizations(); C
Category of graded hopf algebras with basis over Rational Field with realizations
sage: C.super_categories()
[Category of graded hopf algebras over Rational Field]
sage: C.all_super_categories()
[Category of graded hopf algebras with basis over Rational Field with realizations,
 Category of graded hopf algebras over Rational Field,
 Category of graded bialgebras over Rational Field,
 Category of graded algebras over Rational Field,
 ...]

sage: TestSuite(Semigroups().WithRealizations()).run()
class sage.categories.with_realizations.WithRealizationsCategory(category, *args)

Bases: sage.categories.covariant_functorial_construction.RegressiveCovariantConstructionCategory

An abstract base class for all categories of parents with multiple realizations.

The role of this base class is to implement some technical goodies, such as the name for that category.

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