With Realizations Covariant Functorial Construction#

See also

The documentation and code (sage.categories.examples.with_realizations) of Sets().WithRealizations().example() for more on how to use and implement a parent with several realizations.
Various use cases:
- SymmetricFunctions
- QuasiSymmetricFunctions
- NonCommutativeSymmetricFunctions
- SymmetricFunctionsNonCommutingVariables
- DescentAlgebra
- algebras.Moebius
- IwahoriHeckeAlgebra
- ExtendedAffineWeylGroup
The Implementing Algebraic Structures thematic tutorial.
sage.categories.realizations

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

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

Python

>>> from sage.all import *
>>> 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.Facade) admitting multiple concrete realizations where its elements are represented. Consider for example an algebra \(A\) which admits several natural bases:

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

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

sage: # needs sage.modules
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}]

Python

>>> from sage.all import *
>>> # needs sage.modules
>>> A.F()
The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis
>>> A.Out()
The subset algebra of {1, 2, 3} over Rational Field in the Out basis
>>> A.In()
The subset algebra of {1, 2, 3} over Rational Field in the In basis
>>> 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

sage: # needs sage.modules
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

Python

>>> from sage.all import *
>>> # needs sage.modules
>>> F = A.F(); In = A.In(); Out = A.Out()
>>> i = In.an_element(); i
In[{}] + 2*In[{1}] + 3*In[{2}] + In[{1, 2}]
>>> F(i)
7*F[{}] + 3*F[{1}] + 4*F[{2}] + F[{1, 2}]
>>> 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

sage: (1 + Out.from_set(1)) * In.from_set(2,3)                                  # needs sage.modules
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}]

Python

>>> from sage.all import *
>>> (Integer(1) + Out.from_set(Integer(1))) * In.from_set(Integer(2),Integer(3))                                  # needs sage.modules
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

sage: A.realizations()                                                          # needs sage.modules
[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]

Python

>>> from sage.all import *
>>> A.realizations()                                                          # needs sage.modules
[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]

Instead of manually defining the shorthands F, In, and Out, as above one can just do:

Sage

sage: A.inject_shorthands()                                                     # needs sage.modules
Defining F as shorthand for
 The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis
Defining In as shorthand for
 The subset algebra of {1, 2, 3} over Rational Field in the In basis
Defining Out as shorthand for
 The subset algebra of {1, 2, 3} over Rational Field in the Out basis

Python

>>> from sage.all import *
>>> A.inject_shorthands()                                                     # needs sage.modules
Defining F as shorthand for
 The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis
Defining In as shorthand for
 The subset algebra of {1, 2, 3} over Rational Field in the In basis
Defining Out as shorthand for
 The subset algebra of {1, 2, 3} over Rational Field in the Out basis

Rationale

Besides some goodies described below, the role of \(A\) is threefold:

To provide, as illustrated above, a single entry point for the algebra as a whole: documentation, access to its properties and different realizations, etc.
To provide a natural location for the initialization of the bases and the coercions between, and other methods that are common to all bases.
To let other objects refer to \(A\) while allowing elements to be represented in any of the realizations.

We now illustrate this second point by defining the polynomial ring with coefficients in \(A\):

Sage

sage: P = A['x']; P                                                             # needs sage.modules
Univariate Polynomial Ring in x over
 The subset algebra of {1, 2, 3} over Rational Field
sage: x = P.gen()                                                               # needs sage.modules

Python

>>> from sage.all import *
>>> P = A['x']; P                                                             # needs sage.modules
Univariate Polynomial Ring in x over
 The subset algebra of {1, 2, 3} over Rational Field
>>> x = P.gen()                                                               # needs sage.modules

In the following examples, the coefficients turn out to be all represented in the \(F\) basis:

Sage

sage: P.one()                                                                   # needs sage.modules
F[{}]
sage: (P.an_element() + 1)^2                                                    # needs sage.modules
F[{}]*x^2 + 2*F[{}]*x + F[{}]

Python

>>> from sage.all import *
>>> P.one()                                                                   # needs sage.modules
F[{}]
>>> (P.an_element() + Integer(1))**Integer(2)                                                    # needs sage.modules
F[{}]*x^2 + 2*F[{}]*x + F[{}]

However we can create a polynomial with mixed coefficients, and compute with it:

Sage

sage: p = P([1, In[{1}], Out[{2}] ]); p                                         # needs sage.modules
Out[{2}]*x^2 + In[{1}]*x + F[{}]
sage: p^2                                                                       # needs sage.modules
Out[{2}]*x^4
+ (-8*In[{}] + 4*In[{1}] + 8*In[{2}] + 4*In[{3}]
   - 4*In[{1, 2}] - 2*In[{1, 3}] - 4*In[{2, 3}] + 2*In[{1, 2, 3}])*x^3
+ (F[{}] + 3*F[{1}] + 2*F[{2}] - 2*F[{1, 2}] - 2*F[{2, 3}] + 2*F[{1, 2, 3}])*x^2
+ (2*F[{}] + 2*F[{1}])*x
+ F[{}]

Python

>>> from sage.all import *
>>> p = P([Integer(1), In[{Integer(1)}], Out[{Integer(2)}] ]); p                                         # needs sage.modules
Out[{2}]*x^2 + In[{1}]*x + F[{}]
>>> p**Integer(2)                                                                       # needs sage.modules
Out[{2}]*x^4
+ (-8*In[{}] + 4*In[{1}] + 8*In[{2}] + 4*In[{3}]
   - 4*In[{1, 2}] - 2*In[{1, 3}] - 4*In[{2, 3}] + 2*In[{1, 2, 3}])*x^3
+ (F[{}] + 3*F[{1}] + 2*F[{2}] - 2*F[{1, 2}] - 2*F[{2, 3}] + 2*F[{1, 2, 3}])*x^2
+ (2*F[{}] + 2*F[{1}])*x
+ F[{}]

Note how each coefficient involves a single basis which need not be that of the other coefficients. Which basis is used depends on how coercion happened during mixed arithmetic and needs not be deterministic.

One can easily coerce all coefficient to a given basis with:

Sage

sage: p.map_coefficients(In)                                                    # needs sage.modules
(-4*In[{}] + 2*In[{1}] + 4*In[{2}] + 2*In[{3}]
 - 2*In[{1, 2}] - In[{1, 3}] - 2*In[{2, 3}] + In[{1, 2, 3}])*x^2
+ In[{1}]*x + In[{}]

Python

>>> from sage.all import *
>>> p.map_coefficients(In)                                                    # needs sage.modules
(-4*In[{}] + 2*In[{1}] + 4*In[{2}] + 2*In[{3}]
 - 2*In[{1, 2}] - In[{1, 3}] - 2*In[{2, 3}] + In[{1, 2, 3}])*x^2
+ In[{1}]*x + In[{}]

Alas, the natural notation for constructing such polynomials does not yet work:

Sage

sage: In[{1}] * x                                                               # needs sage.modules
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'The subset algebra of {1, 2, 3} over Rational Field in the In basis'
and 'Univariate Polynomial Ring in x over
The subset algebra of {1, 2, 3} over Rational Field'

Python

>>> from sage.all import *
>>> In[{Integer(1)}] * x                                                               # needs sage.modules
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'The subset algebra of {1, 2, 3} over Rational Field in the In basis'
and 'Univariate Polynomial Ring in x over
The subset algebra of {1, 2, 3} over Rational Field'

The category of realizations of \(A\)

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

Sage

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

Python

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

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

Sage

sage: A.F() in A.Realizations()                                                 # needs sage.modules
True

Python

>>> from sage.all import *
>>> A.F() in A.Realizations()                                                 # needs sage.modules
True

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

Sage

sage: A in Algebras(QQ).WithRealizations()                                      # needs sage.modules
True

Python

>>> from sage.all import *
>>> A in Algebras(QQ).WithRealizations()                                      # needs sage.modules
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, VectorSpaces(QQ).FiniteDimensional().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

sage: A in Rings().WithRealizations()                                           # needs sage.modules
True

Python

>>> from sage.all import *
>>> A in Rings().WithRealizations()                                           # needs sage.modules
True

and in the category of algebras (regressiveness):

Sage

sage: A in Algebras(QQ)                                                         # needs sage.modules
True

Python

>>> from sage.all import *
>>> A in Algebras(QQ)                                                         # needs sage.modules
True

Note

For C a category, C.WithRealizations() in fact calls sage.categories.with_realizations.WithRealizations(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

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

Python

>>> from sage.all import *
>>> Algebras(QQ).WithRealizations().super_categories()
[Category of algebras over Rational Field,
 Category of monoids with realizations,
 Category of additive unital additive magmas 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.

class sage.categories.with_realizations.WithRealizationsCategory(category, *args)[source]#

Bases: RegressiveCovariantConstructionCategory

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