Homset categories¶
- class sage.categories.homsets.Homsets[source]¶
Bases:
Category_singletonThe category of all homsets.
EXAMPLES:
sage: from sage.categories.homsets import Homsets sage: Homsets() Category of homsets
>>> from sage.all import * >>> from sage.categories.homsets import Homsets >>> Homsets() Category of homsets
This is a subcategory of
Sets():sage: Homsets().super_categories() [Category of sets]
>>> from sage.all import * >>> Homsets().super_categories() [Category of sets]
By this, we assume that all homsets implemented in Sage are sets, or equivalently that we only implement locally small categories. See Wikipedia article Category_(mathematics).
Issue #17364: every homset category shall be a subcategory of the category of all homsets:
sage: Schemes().Homsets().is_subcategory(Homsets()) True sage: AdditiveMagmas().Homsets().is_subcategory(Homsets()) True sage: AdditiveMagmas().AdditiveUnital().Homsets().is_subcategory(Homsets()) True
>>> from sage.all import * >>> Schemes().Homsets().is_subcategory(Homsets()) True >>> AdditiveMagmas().Homsets().is_subcategory(Homsets()) True >>> AdditiveMagmas().AdditiveUnital().Homsets().is_subcategory(Homsets()) True
This is tested in
HomsetsCategory._test_homsets_category().- class Endset(base_category)[source]¶
Bases:
CategoryWithAxiom_singletonThe category of all endomorphism sets.
This category serves too purposes: making sure that the
Endsetaxiom is implemented in the category where it’s defined, namelyHomsets, and specifying thatEndsetsare monoids.EXAMPLES:
sage: from sage.categories.homsets import Homsets sage: Homsets().Endset() Category of endsets
>>> from sage.all import * >>> from sage.categories.homsets import Homsets >>> Homsets().Endset() Category of endsets
- extra_super_categories()[source]¶
Implement the fact that endsets are monoids.
EXAMPLES:
sage: from sage.categories.homsets import Homsets sage: Homsets().Endset().extra_super_categories() [Category of monoids]
>>> from sage.all import * >>> from sage.categories.homsets import Homsets >>> Homsets().Endset().extra_super_categories() [Category of monoids]
- class ParentMethods[source]¶
Bases:
object- is_endomorphism_set()[source]¶
Return
Trueif the domain and codomain ofselfare the same object.EXAMPLES:
sage: P.<t> = ZZ[] sage: f = P.hom([1/2*t]) sage: f.parent().is_endomorphism_set() False sage: g = P.hom([2*t]) sage: g.parent().is_endomorphism_set() True
>>> from sage.all import * >>> P = ZZ['t']; (t,) = P._first_ngens(1) >>> f = P.hom([Integer(1)/Integer(2)*t]) >>> f.parent().is_endomorphism_set() False >>> g = P.hom([Integer(2)*t]) >>> g.parent().is_endomorphism_set() True
- class SubcategoryMethods[source]¶
Bases:
object- Endset()[source]¶
Return the subcategory of the homsets of
selfthat are endomorphism sets.EXAMPLES:
sage: Sets().Homsets().Endset() Category of endsets of sets sage: Posets().Homsets().Endset() Category of endsets of posets
>>> from sage.all import * >>> Sets().Homsets().Endset() Category of endsets of sets >>> Posets().Homsets().Endset() Category of endsets of posets
- class sage.categories.homsets.HomsetsCategory(category, *args)[source]¶
Bases:
FunctorialConstructionCategory,CategoryWithParameters- base()[source]¶
If this homsets category is subcategory of a category with a base, return that base.
Todo
Is this really useful?
EXAMPLES:
sage: ModulesWithBasis(ZZ).Homsets().base() Integer Ring
>>> from sage.all import * >>> ModulesWithBasis(ZZ).Homsets().base() Integer Ring
- classmethod default_super_categories(category)[source]¶
Return the default super categories of
category.Homsets().INPUT:
cls– the category class for the functor \(F\)category– a category \(Cat\)
OUTPUT: a category
As for the other functorial constructions, if
categoryimplements a nestedHomsetsclass, this method is used in combination withcategory.Homsets().extra_super_categories()to compute the super categories ofcategory.Homsets().EXAMPLES:
If
categoryhas one or more full super categories, then the join of their respective homsets category is returned. In this example, this join consists of a single category:sage: from sage.categories.homsets import HomsetsCategory sage: from sage.categories.additive_groups import AdditiveGroups sage: C = AdditiveGroups() sage: C.full_super_categories() [Category of additive inverse additive unital additive magmas, Category of additive monoids] sage: H = HomsetsCategory.default_super_categories(C); H Category of homsets of additive monoids sage: type(H) <class 'sage.categories.additive_monoids.AdditiveMonoids.Homsets_with_category'>
>>> from sage.all import * >>> from sage.categories.homsets import HomsetsCategory >>> from sage.categories.additive_groups import AdditiveGroups >>> C = AdditiveGroups() >>> C.full_super_categories() [Category of additive inverse additive unital additive magmas, Category of additive monoids] >>> H = HomsetsCategory.default_super_categories(C); H Category of homsets of additive monoids >>> type(H) <class 'sage.categories.additive_monoids.AdditiveMonoids.Homsets_with_category'>
and, given that nothing specific is currently implemented for homsets of additive groups,
His directly the category thereof:sage: C.Homsets() Category of homsets of additive monoids
>>> from sage.all import * >>> C.Homsets() Category of homsets of additive monoids
Similarly for rings: a ring homset is just a homset of unital magmas and additive magmas:
sage: Rings().Homsets() Category of homsets of unital magmas and additive unital additive magmas
>>> from sage.all import * >>> Rings().Homsets() Category of homsets of unital magmas and additive unital additive magmas
Otherwise, if
categoryimplements a nested classHomsets, this method returns the category of all homsets:sage: AdditiveMagmas.Homsets <class 'sage.categories.additive_magmas.AdditiveMagmas.Homsets'> sage: HomsetsCategory.default_super_categories(AdditiveMagmas()) Category of homsets
>>> from sage.all import * >>> AdditiveMagmas.Homsets <class 'sage.categories.additive_magmas.AdditiveMagmas.Homsets'> >>> HomsetsCategory.default_super_categories(AdditiveMagmas()) Category of homsets
which gives one of the super categories of
category.Homsets():sage: AdditiveMagmas().Homsets().super_categories() [Category of additive magmas, Category of homsets] sage: AdditiveMagmas().AdditiveUnital().Homsets().super_categories() [Category of additive unital additive magmas, Category of homsets]
>>> from sage.all import * >>> AdditiveMagmas().Homsets().super_categories() [Category of additive magmas, Category of homsets] >>> AdditiveMagmas().AdditiveUnital().Homsets().super_categories() [Category of additive unital additive magmas, Category of homsets]
the other coming from
category.Homsets().extra_super_categories():sage: AdditiveMagmas().Homsets().extra_super_categories() [Category of additive magmas]
>>> from sage.all import * >>> AdditiveMagmas().Homsets().extra_super_categories() [Category of additive magmas]
Finally, as a last resort, this method returns a stub category modelling the homsets of this category:
sage: hasattr(Posets, "Homsets") False sage: H = HomsetsCategory.default_super_categories(Posets()); H Category of homsets of posets sage: type(H) <class 'sage.categories.homsets.HomsetsOf_with_category'> sage: Posets().Homsets() Category of homsets of posets
>>> from sage.all import * >>> hasattr(Posets, "Homsets") False >>> H = HomsetsCategory.default_super_categories(Posets()); H Category of homsets of posets >>> type(H) <class 'sage.categories.homsets.HomsetsOf_with_category'> >>> Posets().Homsets() Category of homsets of posets
- class sage.categories.homsets.HomsetsOf(category, *args)[source]¶
Bases:
HomsetsCategoryDefault class for homsets of a category.
This is used when a category \(C\) defines some additional structure but not a homset category of its own. Indeed, unlike for covariant functorial constructions, we cannot represent the homset category of \(C\) by just the join of the homset categories of its super categories.
EXAMPLES:
sage: C = (Magmas() & Posets()).Homsets(); C Category of homsets of magmas and posets sage: type(C) <class 'sage.categories.homsets.HomsetsOf_with_category'>
>>> from sage.all import * >>> C = (Magmas() & Posets()).Homsets(); C Category of homsets of magmas and posets >>> type(C) <class 'sage.categories.homsets.HomsetsOf_with_category'>
- super_categories()[source]¶
Return the super categories of
self.A stub homset category admits a single super category, namely the category of all homsets.
EXAMPLES:
sage: C = (Magmas() & Posets()).Homsets(); C Category of homsets of magmas and posets sage: type(C) <class 'sage.categories.homsets.HomsetsOf_with_category'> sage: C.super_categories() [Category of homsets]
>>> from sage.all import * >>> C = (Magmas() & Posets()).Homsets(); C Category of homsets of magmas and posets >>> type(C) <class 'sage.categories.homsets.HomsetsOf_with_category'> >>> C.super_categories() [Category of homsets]