Algebra Functorial Construction

AUTHORS:

  • Nicolas M. Thiery (2010): initial revision
class sage.categories.algebra_functor.AlgebraFunctor(base_ring)

Bases: sage.categories.covariant_functorial_construction.CovariantFunctorialConstruction

A singleton class for the algebra functor

base_ring()

Returns the base ring for this functor

EXAMPLES:

sage: from sage.categories.algebra_functor import AlgebraFunctor
sage: AlgebraFunctor(QQ).base_ring()
Rational Field
sage.categories.algebra_functor.Algebras(self, base_ring)

INPUT:

  • self – a subcategory of Sets()
  • base_ring – a ring

Returns the category of objects constructed as algebras of objects of self over base_ring.

EXAMPLES:

sage: Monoids().Algebras(QQ)
Category of monoid algebras over Rational Field

sage: Groups().Algebras(QQ)
Category of group algebras over Rational Field

sage: M = Monoids().example(); M
An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd')
sage: A = M.algebra(QQ); A
Free module generated by An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd') over Rational Field
sage: A.category()
Category of monoid algebras over Rational Field
class sage.categories.algebra_functor.AlgebrasCategory(base_category, base_ring)

Bases: sage.categories.covariant_functorial_construction.CovariantConstructionCategory, sage.categories.category_types.Category_over_base_ring

Returns the category of base_ring-algebras over self

A category with algebra functor is a category endowed with an algebra functor from itself to the category of modules, mapping a set \(S\) and a field \(C\) to a \(C\)-free module with basis indexed by \(S\), more often than not endowed with an algebra structure. Typical examples are the functor from monoids to monoid algebras, groups to group algebras, etc.

See also CovariantFunctorialConstruction.

INPUT:

  • base_ring – a ring

EXAMPLES:

sage: Monoids().Algebras(QQ)
Category of monoid algebras over Rational Field

Previous topic

Dual functorial construction

Next topic

Subquotient Functorial Construction

This Page