AUTHORS:
- Nicolas M. Thiery (2010): initial revision
Bases: sage.categories.covariant_functorial_construction.RegressiveCovariantConstructionCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
sage: class FooBars(CovariantConstructionCategory):
... _functor_category = "FooBars"
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()
Returns the default super categories of category.IsomorphicObjects()
Mathematical meaning: if is the image of
by an
isomorphism in the category
, then
is both a subobject
of
and a quotient of
in the category
.
INPUT:
- cls – the class IsomorphicObjectsCategory
- category – a category
OUTPUT: a (join) category
In practice, this returns category.Subobjects() and category.Quotients(), joined together with the result of the method RegressiveCovariantConstructionCategory.default_super_categories() (that is the join of category and cat.IsomorphicObjects() for each cat in the super categories of category).
EXAMPLES:
Consider category=Groups(), which has cat=Monoids() as
super category. Then, the image of a group by a group
isomorphism is simultaneously a subgroup of
, a subquotient
of
, a group by itself, and the image of
by a monoid
isomorphism:
sage: Groups().IsomorphicObjects().super_categories()
[Category of groups,
Category of subquotients of monoids,
Category of quotients of semigroups,
Category of isomorphic objects of sets]
Mind the last item above: there is indeed currently nothing implemented about isomorphic objects of monoids.
This resulted from the following call:
sage: sage.categories.isomorphic_objects.IsomorphicObjectsCategory.default_super_categories(Groups())
Join of Category of groups and
Category of subquotients of monoids and
Category of quotients of semigroups and
Category of isomorphic objects of sets