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 HomCategory(category, name=None)

Bases: sage.categories.category.HomCategory

Initializes this HomCategory

INPUT:
  • category – the category whose Homsets are the objects of this category.
  • name – An optional name for this category.

EXAMPLES:

We need to skip one test, since the hierarchy of hom categories isn’t consistent yet:

sage: C = sage.categories.category.HomCategory(Rings()); C
Category of hom sets in Category of rings
sage: TestSuite(C).run(skip=['_test_category_graph'])
extra_super_categories()

EXAMPLES:

sage: Schemes().hom_category().extra_super_categories()
[]
sage: Schemes().hom_category().super_categories()
[Category of hom sets in Category of sets]

FIXME: what category structure is there on Homsets of schemes? The result above is wrong, and should be fixed during the next homsets overhaul.

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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