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# Realizations Covariant Functorial Construction

See also

- `Sets().WithRealizations` for an introduction to realizations and with realizations.
- `sage.categories.covariant_functorial_construction` for an introduction to covariant functorial constructions.
- `sage.categories.examples.with_realizations` for an example.

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

  See also

  `Sets().WithRealizations`

  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

  - `Sets().WithRealizations`
  - `ClasscallMetaclass`

  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.

  See also

  `Sets().WithRealizations`

  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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