# Quotients Functorial Construction¶

AUTHORS:

• Nicolas M. Thiery (2010): initial revision
class sage.categories.quotients.QuotientsCategory(category, *args)
classmethod default_super_categories(category)

Returns the default super categories of category.Quotients()

Mathematical meaning: if $$A$$ is a quotient of $$B$$ in the category $$C$$, then $$A$$ is also a subquotient of $$B$$ in the category $$C$$.

INPUT:

• cls – the class QuotientsCategory
• category – a category $$Cat$$

OUTPUT: a (join) category

In practice, this returns category.Subquotients(), joined together with the result of the method RegressiveCovariantConstructionCategory.default_super_categories() (that is the join of category and cat.Quotients() for each cat in the super categories of category).

EXAMPLES:

Consider category=Groups(), which has cat=Monoids() as super category. Then, a subgroup of a group $$G$$ is simultaneously a subquotient of $$G$$, a group by itself, and a quotient monoid of G:

sage: Groups().Quotients().super_categories()
[Category of groups, Category of subquotients of monoids, Category of quotients of semigroups]


Mind the last item above: there is indeed currently nothing implemented about quotient monoids.

This resulted from the following call:

sage: sage.categories.quotients.QuotientsCategory.default_super_categories(Groups())
Join of Category of groups and Category of subquotients of monoids and Category of quotients of semigroups