Schemes

class sage.categories.schemes.Schemes(s=None)

Bases: sage.categories.category.Category

The category of all schemes.

EXAMPLES:

sage: Schemes()
Category of schemes

Schemes can also be used to construct the category of schemes over a given base:

sage: Schemes(Spec(ZZ))
Category of schemes over Integer Ring

sage: Schemes(ZZ)
Category of schemes over Integer Ring

Todo

Make Schemes() a singleton category (and remove Schemes from the workaround in category_types.Category_over_base._test_category_over_bases()).

This is currently incompatible with the dispatching below.

TESTS:

sage: TestSuite(Schemes()).run()
class Homsets(category, *args)

Bases: sage.categories.homsets.HomsetsCategory

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()
extra_super_categories()

EXAMPLES:

sage: Schemes().Homsets().extra_super_categories()
[]
sage: Schemes().Homsets().super_categories()
[Category of objects]

Todo

What category structure is there on Homsets of schemes?

Todo

check that the result above is correct now

Schemes.super_categories()

EXAMPLES:

sage: Schemes().super_categories()
[Category of sets]
class sage.categories.schemes.Schemes_over_base(base, name=None)

Bases: sage.categories.category_types.Category_over_base

The category of schemes over a given base scheme.

EXAMPLES:

sage: Schemes(Spec(ZZ))
Category of schemes over Integer Ring

TESTS:

sage: C = Schemes(ZZ)
sage: TestSuite(C).run()
base_scheme()

EXAMPLES:

sage: Schemes(Spec(ZZ)).base_scheme()
Spectrum of Integer Ring
super_categories()

EXAMPLES:

sage: Schemes(Spec(ZZ)).super_categories()
[Category of schemes]

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