Subobjects Functorial Construction

AUTHORS:

  • Nicolas M. Thiery (2010): initial revision
sage.categories.subobjects.Subobjects(self)
INPUT:
  • self – a concrete category

Given a concrete category As() (i.e. a subcategory of Sets()), As().Subobjects() returns the category of objects of As() endowed with a distinguished description as subobject of some other object of As().

See Subquotients() for background.

EXAMPLES:

sage: C = Sets().Subobjects(); C
Category of subobjects of sets

sage: C.super_categories()
[Category of subquotients of sets]

sage: C.all_super_categories()
[Category of subobjects of sets,
 Category of subquotients of sets,
 Category of sets,
 Category of sets with partial maps,
 Category of objects]

Unless something specific about subobjects is implemented for this category, one actually get an optimized super category:

sage: C = Semigroups().Subobjects(); C
Join of Category of subquotients of semigroups and Category of subobjects of sets

The caller is responsible for checking that the given category admits a well defined category of subobjects.

TESTS:

sage: Semigroups().Subobjects().is_subcategory(Semigroups().Subquotients())
True
sage: TestSuite(C).run()
class sage.categories.subobjects.SubobjectsCategory(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"
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()
classmethod default_super_categories(category)

Returns the default super categories of category.Subobjects()

Mathematical meaning: if \(A\) is a subobject of \(B\) in the category \(C\), then \(A\) is also a subquotient of \(B\) in the category \(C\).

INPUT:

  • cls – the class SubobjectsCategory
  • 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.Subobjects() 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 submonoid of \(G\):

sage: Groups().Subobjects().super_categories()
[Category of groups, Category of subquotients of monoids, Category of subobjects of sets]

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

This resulted from the following call:

sage: sage.categories.subobjects.SubobjectsCategory.default_super_categories(Groups())
Join of Category of groups and Category of subquotients of monoids and Category of subobjects of sets

Previous topic

Quotients Functorial Construction

Next topic

Isomorphic Objects Functorial Construction

This Page