# Algebra Functorial Construction¶

AUTHORS:

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

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)

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.

INPUT:

• base_ring – a ring

EXAMPLES:

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


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