Functors#
AUTHORS:
David Kohel and William Stein
David Joyner (2005-12-17): examples
Robert Bradshaw (2007-06-23): Pyrexify
Simon King (2010-04-30): more examples, several bug fixes, re-implementation of the default call method, making functors applicable to morphisms (not only to objects)
Simon King (2010-12): Pickling of functors without losing domain and codomain
- sage.categories.functor.ForgetfulFunctor(domain, codomain)[source]#
Construct the forgetful function from one category to another.
INPUT:
C
,D
– two categoriesOUTPUT:
A functor that returns the corresponding object of
D
for any element ofC
, by forgetting the extra structure.ASSUMPTION:
The category
C
must be a sub-category ofD
.EXAMPLES:
sage: rings = Rings() sage: abgrps = CommutativeAdditiveGroups() sage: F = ForgetfulFunctor(rings, abgrps) sage: F The forgetful functor from Category of rings to Category of commutative additive groups
>>> from sage.all import * >>> rings = Rings() >>> abgrps = CommutativeAdditiveGroups() >>> F = ForgetfulFunctor(rings, abgrps) >>> F The forgetful functor from Category of rings to Category of commutative additive groups
It would be a mistake to call it in opposite order:
sage: F = ForgetfulFunctor(abgrps, rings) Traceback (most recent call last): ... ValueError: Forgetful functor not supported for domain Category of commutative additive groups
>>> from sage.all import * >>> F = ForgetfulFunctor(abgrps, rings) Traceback (most recent call last): ... ValueError: Forgetful functor not supported for domain Category of commutative additive groups
If both categories are equal, the forgetful functor is the same as the identity functor:
sage: ForgetfulFunctor(abgrps, abgrps) == IdentityFunctor(abgrps) True
>>> from sage.all import * >>> ForgetfulFunctor(abgrps, abgrps) == IdentityFunctor(abgrps) True
- class sage.categories.functor.ForgetfulFunctor_generic[source]#
Bases:
Functor
The forgetful functor, i.e., embedding of a subcategory.
NOTE:
Forgetful functors should be created using
ForgetfulFunctor()
, since the init method of this class does not check whether the domain is a subcategory of the codomain.EXAMPLES:
sage: F = ForgetfulFunctor(FiniteFields(), Fields()) # indirect doctest sage: F The forgetful functor from Category of finite enumerated fields to Category of fields sage: F(GF(3)) Finite Field of size 3
>>> from sage.all import * >>> F = ForgetfulFunctor(FiniteFields(), Fields()) # indirect doctest >>> F The forgetful functor from Category of finite enumerated fields to Category of fields >>> F(GF(Integer(3))) Finite Field of size 3
- class sage.categories.functor.Functor[source]#
Bases:
SageObject
A class for functors between two categories
NOTE:
In the first place, a functor is given by its domain and codomain, which are both categories.
When defining a sub-class, the user should not implement a call method. Instead, one should implement three methods, which are composed in the default call method:
_coerce_into_domain(self, x)
: Return an object ofself
’s domain, corresponding tox
, or raise aTypeError
.Default: Raise
TypeError
ifx
is not inself
’s domain.
_apply_functor(self, x)
: Applyself
to an objectx
ofself
’s domain.Default: Conversion into
self
’s codomain.
_apply_functor_to_morphism(self, f)
: Applyself
to a morphismf
inself
’s domain. - Default: Returnself(f.domain()).hom(f,self(f.codomain()))
.
EXAMPLES:
sage: rings = Rings() sage: abgrps = CommutativeAdditiveGroups() sage: F = ForgetfulFunctor(rings, abgrps) sage: F.domain() Category of rings sage: F.codomain() Category of commutative additive groups sage: from sage.categories.functor import Functor sage: isinstance(F, Functor) True sage: I = IdentityFunctor(abgrps) sage: I The identity functor on Category of commutative additive groups sage: I.domain() Category of commutative additive groups sage: isinstance(I, Functor) True
>>> from sage.all import * >>> rings = Rings() >>> abgrps = CommutativeAdditiveGroups() >>> F = ForgetfulFunctor(rings, abgrps) >>> F.domain() Category of rings >>> F.codomain() Category of commutative additive groups >>> from sage.categories.functor import Functor >>> isinstance(F, Functor) True >>> I = IdentityFunctor(abgrps) >>> I The identity functor on Category of commutative additive groups >>> I.domain() Category of commutative additive groups >>> isinstance(I, Functor) True
Note that by default, an instance of the class Functor is coercion from the domain into the codomain. The above subclasses overloaded this behaviour. Here we illustrate the default:
sage: from sage.categories.functor import Functor sage: F = Functor(Rings(), Fields()) sage: F Functor from Category of rings to Category of fields sage: F(ZZ) Rational Field sage: F(GF(2)) Finite Field of size 2
>>> from sage.all import * >>> from sage.categories.functor import Functor >>> F = Functor(Rings(), Fields()) >>> F Functor from Category of rings to Category of fields >>> F(ZZ) Rational Field >>> F(GF(Integer(2))) Finite Field of size 2
Functors are not only about the objects of a category, but also about their morphisms. We illustrate it, again, with the coercion functor from rings to fields.
sage: R1.<x> = ZZ[] sage: R2.<a,b> = QQ[] sage: f = R1.hom([a + b], R2) sage: f Ring morphism: From: Univariate Polynomial Ring in x over Integer Ring To: Multivariate Polynomial Ring in a, b over Rational Field Defn: x |--> a + b sage: F(f) Ring morphism: From: Fraction Field of Univariate Polynomial Ring in x over Integer Ring To: Fraction Field of Multivariate Polynomial Ring in a, b over Rational Field Defn: x |--> a + b sage: F(f)(1/x) 1/(a + b)
>>> from sage.all import * >>> R1 = ZZ['x']; (x,) = R1._first_ngens(1) >>> R2 = QQ['a, b']; (a, b,) = R2._first_ngens(2) >>> f = R1.hom([a + b], R2) >>> f Ring morphism: From: Univariate Polynomial Ring in x over Integer Ring To: Multivariate Polynomial Ring in a, b over Rational Field Defn: x |--> a + b >>> F(f) Ring morphism: From: Fraction Field of Univariate Polynomial Ring in x over Integer Ring To: Fraction Field of Multivariate Polynomial Ring in a, b over Rational Field Defn: x |--> a + b >>> F(f)(Integer(1)/x) 1/(a + b)
We can also apply a polynomial ring construction functor to our homomorphism. The result is a homomorphism that is defined on the base ring:
sage: F = QQ['t'].construction()[0] sage: F Poly[t] sage: F(f) Ring morphism: From: Univariate Polynomial Ring in t over Univariate Polynomial Ring in x over Integer Ring To: Univariate Polynomial Ring in t over Multivariate Polynomial Ring in a, b over Rational Field Defn: Induced from base ring by Ring morphism: From: Univariate Polynomial Ring in x over Integer Ring To: Multivariate Polynomial Ring in a, b over Rational Field Defn: x |--> a + b sage: p = R1['t']('(-x^2 + x)*t^2 + (x^2 - x)*t - 4*x^2 - x + 1') sage: F(f)(p) (-a^2 - 2*a*b - b^2 + a + b)*t^2 + (a^2 + 2*a*b + b^2 - a - b)*t - 4*a^2 - 8*a*b - 4*b^2 - a - b + 1
>>> from sage.all import * >>> F = QQ['t'].construction()[Integer(0)] >>> F Poly[t] >>> F(f) Ring morphism: From: Univariate Polynomial Ring in t over Univariate Polynomial Ring in x over Integer Ring To: Univariate Polynomial Ring in t over Multivariate Polynomial Ring in a, b over Rational Field Defn: Induced from base ring by Ring morphism: From: Univariate Polynomial Ring in x over Integer Ring To: Multivariate Polynomial Ring in a, b over Rational Field Defn: x |--> a + b >>> p = R1['t']('(-x^2 + x)*t^2 + (x^2 - x)*t - 4*x^2 - x + 1') >>> F(f)(p) (-a^2 - 2*a*b - b^2 + a + b)*t^2 + (a^2 + 2*a*b + b^2 - a - b)*t - 4*a^2 - 8*a*b - 4*b^2 - a - b + 1
- sage.categories.functor.IdentityFunctor(C)[source]#
Construct the identity functor of the given category.
INPUT:
A category,
C
.OUTPUT:
The identity functor in
C
.EXAMPLES:
sage: rings = Rings() sage: F = IdentityFunctor(rings) sage: F(ZZ['x','y']) is ZZ['x','y'] True
>>> from sage.all import * >>> rings = Rings() >>> F = IdentityFunctor(rings) >>> F(ZZ['x','y']) is ZZ['x','y'] True
- class sage.categories.functor.IdentityFunctor_generic(C)[source]#
Bases:
ForgetfulFunctor_generic
Generic identity functor on any category
NOTE:
This usually is created using
IdentityFunctor()
.EXAMPLES:
sage: F = IdentityFunctor(Fields()) #indirect doctest sage: F The identity functor on Category of fields sage: F(RR) is RR True sage: F(ZZ) Traceback (most recent call last): ... TypeError: x (=Integer Ring) is not in Category of fields
>>> from sage.all import * >>> F = IdentityFunctor(Fields()) #indirect doctest >>> F The identity functor on Category of fields >>> F(RR) is RR True >>> F(ZZ) Traceback (most recent call last): ... TypeError: x (=Integer Ring) is not in Category of fields
- sage.categories.functor.is_Functor(x)[source]#
Test whether the argument is a functor.
This function is deprecated.
EXAMPLES:
sage: from sage.categories.functor import is_Functor sage: F1 = QQ.construction()[0] sage: F1 FractionField sage: is_Functor(F1) doctest:warning... DeprecationWarning: The function is_Functor is deprecated; use 'isinstance(..., Functor)' instead. See https://github.com/sagemath/sage/issues/38184 for details. True sage: is_Functor(FractionField) False sage: F2 = ForgetfulFunctor(Fields(), Rings()) sage: F2 The forgetful functor from Category of fields to Category of rings sage: is_Functor(F2) True
>>> from sage.all import * >>> from sage.categories.functor import is_Functor >>> F1 = QQ.construction()[Integer(0)] >>> F1 FractionField >>> is_Functor(F1) doctest:warning... DeprecationWarning: The function is_Functor is deprecated; use 'isinstance(..., Functor)' instead. See https://github.com/sagemath/sage/issues/38184 for details. True >>> is_Functor(FractionField) False >>> F2 = ForgetfulFunctor(Fields(), Rings()) >>> F2 The forgetful functor from Category of fields to Category of rings >>> is_Functor(F2) True