Quotients Functorial Construction#
AUTHORS:
Nicolas M. Thiery (2010): initial revision
- class sage.categories.quotients.QuotientsCategory(category, *args)[source]#
Bases:
RegressiveCovariantConstructionCategory
- classmethod default_super_categories(category)[source]#
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 classQuotientsCategory
category
– a category \(Cat\)
OUTPUT: a (join) category
In practice, this returns
category.Subquotients()
, joined together with the result of the methodRegressiveCovariantConstructionCategory.default_super_categories()
(that is the join ofcategory
andcat.Quotients()
for eachcat
in the super categories ofcategory
).EXAMPLES:
Consider
category=Groups()
, which hascat=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 ofG
:sage: Groups().Quotients().super_categories() [Category of groups, Category of subquotients of monoids, Category of quotients of semigroups]
>>> from sage.all import * >>> 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
>>> from sage.all import * >>> 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