Homset categories#

class sage.categories.homsets.Homsets[source]#

Bases: Category_singleton

The category of all homsets.

EXAMPLES:

Sage

sage: from sage.categories.homsets import Homsets
sage: Homsets()
Category of homsets

Python

>>> from sage.all import *
>>> from sage.categories.homsets import Homsets
>>> Homsets()
Category of homsets

This is a subcategory of Sets():

Sage

sage: Homsets().super_categories()
[Category of sets]

Python

>>> 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

sage: Schemes().Homsets().is_subcategory(Homsets())
True
sage: AdditiveMagmas().Homsets().is_subcategory(Homsets())
True
sage: AdditiveMagmas().AdditiveUnital().Homsets().is_subcategory(Homsets())
True

Python

>>> 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_singleton

The category of all endomorphism sets.

This category serves too purposes: making sure that the Endset axiom is implemented in the category where it’s defined, namely Homsets, and specifying that Endsets are monoids.

EXAMPLES:

Sage

sage: from sage.categories.homsets import Homsets
sage: Homsets().Endset()
Category of endsets

Python

>>> from sage.all import *
>>> from sage.categories.homsets import Homsets
>>> Homsets().Endset()
Category of endsets

class ParentMethods[source]#

Bases: object

is_endomorphism_set()[source]#

Return True as self is in the category of Endsets.

EXAMPLES:

Sage

sage: P.<t> = ZZ[]
sage: E = End(P)
sage: E.is_endomorphism_set()
True

Python

>>> from sage.all import *
>>> P = ZZ['t']; (t,) = P._first_ngens(1)
>>> E = End(P)
>>> E.is_endomorphism_set()
True

extra_super_categories()[source]#

Implement the fact that endsets are monoids.

EXAMPLES:

Sage

sage: from sage.categories.homsets import Homsets
sage: Homsets().Endset().extra_super_categories()
[Category of monoids]

Python

>>> 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 True if the domain and codomain of self are the same object.

EXAMPLES:

Sage

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

Python

>>> 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 self that are endomorphism sets.

EXAMPLES:

Sage

sage: Sets().Homsets().Endset()
Category of endsets of sets

sage: Posets().Homsets().Endset()
Category of endsets of posets

Python

>>> from sage.all import *
>>> Sets().Homsets().Endset()
Category of endsets of sets

>>> Posets().Homsets().Endset()
Category of endsets of posets

super_categories()[source]#

Return the super categories of self.

EXAMPLES:

Sage

sage: from sage.categories.homsets import Homsets
sage: Homsets()
Category of homsets

Python

>>> from sage.all import *
>>> from sage.categories.homsets import Homsets
>>> Homsets()
Category of homsets

class sage.categories.homsets.HomsetsCategory(category, *args)[source]#

Bases: FunctorialConstructionCategory

base()[source]#

If this homsets category is subcategory of a category with a base, return that base.

Todo

Is this really useful?

EXAMPLES:

Sage

sage: ModulesWithBasis(ZZ).Homsets().base()
Integer Ring

Python

>>> 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 category implements a nested Homsets class, this method is used in combination with category.Homsets().extra_super_categories() to compute the super categories of category.Homsets().

EXAMPLES:

If category has 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

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'>

Python

>>> 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, H is directly the category thereof:

Sage

sage: C.Homsets()
Category of homsets of additive monoids

Python

>>> 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

sage: Rings().Homsets()
Category of homsets of unital magmas and additive unital additive magmas

Python

>>> from sage.all import *
>>> Rings().Homsets()
Category of homsets of unital magmas and additive unital additive magmas

Otherwise, if category implements a nested class Homsets, this method returns the category of all homsets:

Sage

sage: AdditiveMagmas.Homsets
<class 'sage.categories.additive_magmas.AdditiveMagmas.Homsets'>
sage: HomsetsCategory.default_super_categories(AdditiveMagmas())
Category of homsets

Python

>>> 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

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]

Python

>>> 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

sage: AdditiveMagmas().Homsets().extra_super_categories()
[Category of additive magmas]

Python

>>> 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

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

Python

>>> 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: HomsetsCategory

Default 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

sage: C = (Magmas() & Posets()).Homsets(); C
Category of homsets of magmas and posets
sage: type(C)
<class 'sage.categories.homsets.HomsetsOf_with_category'>

Python

>>> 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

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]

Python

>>> 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]