Quotients Functorial Construction¶
AUTHORS:
- Nicolas M. Thiery (2010): initial revision
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class
sage.categories.quotients.
QuotientsCategory
(category, *args)¶ Bases:
sage.categories.covariant_functorial_construction.RegressiveCovariantConstructionCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory sage: class FooBars(CovariantConstructionCategory): ....: _functor_category = "FooBars" ....: _base_category_class = (Category,) sage: Category.FooBars = lambda self: FooBars.category_of(self) sage: C = FooBars(ModulesWithBasis(ZZ)) sage: C Category of foo bars of modules with basis over Integer Ring sage: C.base_category() Category of modules with basis over Integer Ring sage: latex(C) \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}}) sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module sage: TestSuite(C).run()
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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 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]
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
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classmethod