Morphisms#

This module defines the base classes of morphisms between objects of a given category.

EXAMPLES:

Typically, a morphism is defined by the images of the generators of the domain.

sage: X.<a, b> = ZZ[]
sage: Y.<c> = ZZ[]
sage: X.hom([c, c^2])
Ring morphism:
  From: Multivariate Polynomial Ring in a, b over Integer Ring
  To:   Univariate Polynomial Ring in c over Integer Ring
  Defn: a |--> c
        b |--> c^2
>>> from sage.all import *
>>> X = ZZ['a, b']; (a, b,) = X._first_ngens(2)
>>> Y = ZZ['c']; (c,) = Y._first_ngens(1)
>>> X.hom([c, c**Integer(2)])
Ring morphism:
  From: Multivariate Polynomial Ring in a, b over Integer Ring
  To:   Univariate Polynomial Ring in c over Integer Ring
  Defn: a |--> c
        b |--> c^2

AUTHORS:

  • William Stein (2005): initial version

  • David Joyner (2005-12-17): added examples

  • Robert Bradshaw (2007-06-25): Pyrexification

class sage.categories.morphism.CallMorphism[source]#

Bases: Morphism

class sage.categories.morphism.FormalCoercionMorphism[source]#

Bases: Morphism

class sage.categories.morphism.IdentityMorphism[source]#

Bases: Morphism

is_identity()[source]#

Return True if this morphism is the identity morphism.

EXAMPLES:

sage: E = End(Partitions(5))                                                # needs sage.combinat
sage: E.identity().is_identity()                                            # needs sage.combinat
True
>>> from sage.all import *
>>> E = End(Partitions(Integer(5)))                                                # needs sage.combinat
>>> E.identity().is_identity()                                            # needs sage.combinat
True

Check that Issue #15478 is fixed:

sage: # needs sage.rings.finite_rings
sage: K.<z> = GF(4)
sage: phi = End(K)([z^2])
sage: R.<t> = K[]
sage: psi = End(R)(phi)
sage: psi.is_identity()
False
>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> K = GF(Integer(4), names=('z',)); (z,) = K._first_ngens(1)
>>> phi = End(K)([z**Integer(2)])
>>> R = K['t']; (t,) = R._first_ngens(1)
>>> psi = End(R)(phi)
>>> psi.is_identity()
False
is_injective()[source]#

Return whether this morphism is injective.

EXAMPLES:

sage: Hom(ZZ, ZZ).identity().is_injective()
True
>>> from sage.all import *
>>> Hom(ZZ, ZZ).identity().is_injective()
True
is_surjective()[source]#

Return whether this morphism is surjective.

EXAMPLES:

sage: Hom(ZZ, ZZ).identity().is_surjective()
True
>>> from sage.all import *
>>> Hom(ZZ, ZZ).identity().is_surjective()
True
section()[source]#

Return a section of this morphism.

EXAMPLES:

sage: T = Hom(ZZ, ZZ).identity()
sage: T.section() is T
True
>>> from sage.all import *
>>> T = Hom(ZZ, ZZ).identity()
>>> T.section() is T
True
class sage.categories.morphism.Morphism[source]#

Bases: Map

category()[source]#

Return the category of the parent of this morphism.

EXAMPLES:

sage: R.<t> = ZZ[]
sage: f = R.hom([t**2])
sage: f.category()
Category of endsets of unital magmas and right modules over
 (Dedekind domains and euclidean domains
  and noetherian rings
  and infinite enumerated sets and metric spaces)
 and left modules over
 (Dedekind domains and euclidean domains
  and noetherian rings
  and infinite enumerated sets and metric spaces)

sage: # needs sage.rings.number_field
sage: K = CyclotomicField(12)
sage: L = CyclotomicField(132)
sage: phi = L._internal_coerce_map_from(K)
sage: phi.category()
Category of homsets of number fields
>>> from sage.all import *
>>> R = ZZ['t']; (t,) = R._first_ngens(1)
>>> f = R.hom([t**Integer(2)])
>>> f.category()
Category of endsets of unital magmas and right modules over
 (Dedekind domains and euclidean domains
  and noetherian rings
  and infinite enumerated sets and metric spaces)
 and left modules over
 (Dedekind domains and euclidean domains
  and noetherian rings
  and infinite enumerated sets and metric spaces)

>>> # needs sage.rings.number_field
>>> K = CyclotomicField(Integer(12))
>>> L = CyclotomicField(Integer(132))
>>> phi = L._internal_coerce_map_from(K)
>>> phi.category()
Category of homsets of number fields
is_endomorphism()[source]#

Return True if this morphism is an endomorphism.

EXAMPLES:

sage: R.<t> = ZZ[]
sage: f = R.hom([t])
sage: f.is_endomorphism()
True

sage: # needs sage.rings.number_field
sage: K = CyclotomicField(12)
sage: L = CyclotomicField(132)
sage: phi = L._internal_coerce_map_from(K)
sage: phi.is_endomorphism()
False
>>> from sage.all import *
>>> R = ZZ['t']; (t,) = R._first_ngens(1)
>>> f = R.hom([t])
>>> f.is_endomorphism()
True

>>> # needs sage.rings.number_field
>>> K = CyclotomicField(Integer(12))
>>> L = CyclotomicField(Integer(132))
>>> phi = L._internal_coerce_map_from(K)
>>> phi.is_endomorphism()
False
is_identity()[source]#

Return True if this morphism is the identity morphism.

Note

Implemented only when the domain has a method gens()

EXAMPLES:

sage: R.<t> = ZZ[]
sage: f = R.hom([t])
sage: f.is_identity()
True
sage: g = R.hom([t + 1])
sage: g.is_identity()
False
>>> from sage.all import *
>>> R = ZZ['t']; (t,) = R._first_ngens(1)
>>> f = R.hom([t])
>>> f.is_identity()
True
>>> g = R.hom([t + Integer(1)])
>>> g.is_identity()
False

A morphism between two different spaces cannot be the identity:

sage: R2.<t2> = QQ[]
sage: h = R.hom([t2])
sage: h.is_identity()
False
>>> from sage.all import *
>>> R2 = QQ['t2']; (t2,) = R2._first_ngens(1)
>>> h = R.hom([t2])
>>> h.is_identity()
False
pushforward(I)[source]#
register_as_coercion()[source]#

Register this morphism as a coercion to Sage’s coercion model (see sage.structure.coerce).

EXAMPLES:

By default, adding polynomials over different variables triggers an error:

sage: X.<x> = ZZ[]
sage: Y.<y> = ZZ[]
sage: x^2 + y
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +:
'Univariate Polynomial Ring in x over Integer Ring' and
'Univariate Polynomial Ring in y over Integer Ring'
>>> from sage.all import *
>>> X = ZZ['x']; (x,) = X._first_ngens(1)
>>> Y = ZZ['y']; (y,) = Y._first_ngens(1)
>>> x**Integer(2) + y
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +:
'Univariate Polynomial Ring in x over Integer Ring' and
'Univariate Polynomial Ring in y over Integer Ring'

Let us declare a coercion from \(\ZZ[x]\) to \(\ZZ[z]\):

sage: Z.<z> = ZZ[]
sage: phi = Hom(X, Z)(z)
sage: phi(x^2+1)
z^2 + 1
sage: phi.register_as_coercion()
>>> from sage.all import *
>>> Z = ZZ['z']; (z,) = Z._first_ngens(1)
>>> phi = Hom(X, Z)(z)
>>> phi(x**Integer(2)+Integer(1))
z^2 + 1
>>> phi.register_as_coercion()

Now we can add elements from \(\ZZ[x]\) and \(\ZZ[z]\), because the elements of the former are allowed to be implicitly coerced into the later:

sage: x^2 + z
z^2 + z
>>> from sage.all import *
>>> x**Integer(2) + z
z^2 + z

Caveat: the registration of the coercion must be done before any other coercion is registered or discovered:

sage: phi = Hom(X, Z)(z^2)
sage: phi.register_as_coercion()
Traceback (most recent call last):
...
AssertionError: coercion from Univariate Polynomial Ring in x over Integer Ring
to Univariate Polynomial Ring in z over Integer Ring
already registered or discovered
>>> from sage.all import *
>>> phi = Hom(X, Z)(z**Integer(2))
>>> phi.register_as_coercion()
Traceback (most recent call last):
...
AssertionError: coercion from Univariate Polynomial Ring in x over Integer Ring
to Univariate Polynomial Ring in z over Integer Ring
already registered or discovered
register_as_conversion()[source]#

Register this morphism as a conversion to Sage’s coercion model

(see sage.structure.coerce).

EXAMPLES:

Let us declare a conversion from the symmetric group to \(\ZZ\) through the sign map:

sage: # needs sage.groups
sage: S = SymmetricGroup(4)
sage: phi = Hom(S, ZZ)(lambda x: ZZ(x.sign()))
sage: x = S.an_element(); x
(2,3,4)
sage: phi(x)
1
sage: phi.register_as_conversion()
sage: ZZ(x)
1
>>> from sage.all import *
>>> # needs sage.groups
>>> S = SymmetricGroup(Integer(4))
>>> phi = Hom(S, ZZ)(lambda x: ZZ(x.sign()))
>>> x = S.an_element(); x
(2,3,4)
>>> phi(x)
1
>>> phi.register_as_conversion()
>>> ZZ(x)
1
class sage.categories.morphism.SetIsomorphism[source]#

Bases: SetMorphism

An isomorphism of sets.

INPUT:

  • parent – a Homset

  • function – a Python function that takes elements of the domain as input and returns elements of the codomain.

EXAMPLES:

sage: f = sage.categories.morphism.SetIsomorphism(Hom(ZZ, ZZ, Sets()),
....:                                             operator.__neg__); f
Generic endomorphism of Integer Ring
sage: f._set_inverse(f)
sage: ~f is f
True
>>> from sage.all import *
>>> f = sage.categories.morphism.SetIsomorphism(Hom(ZZ, ZZ, Sets()),
...                                             operator.__neg__); f
Generic endomorphism of Integer Ring
>>> f._set_inverse(f)
>>> ~f is f
True
is_injective()[source]#

Return whether this morphism is injective.

EXAMPLES:

sage: f = sage.categories.morphism.SetIsomorphism(Hom(ZZ, ZZ, Sets()),
....:                                             operator.__neg__)
sage: f.is_injective()
True
>>> from sage.all import *
>>> f = sage.categories.morphism.SetIsomorphism(Hom(ZZ, ZZ, Sets()),
...                                             operator.__neg__)
>>> f.is_injective()
True
is_surjective()[source]#

Return whether this morphism is surjective.

EXAMPLES:

sage: f = sage.categories.morphism.SetIsomorphism(Hom(ZZ, ZZ, Sets()),
....:                                             operator.__neg__)
sage: f.is_surjective()
True
>>> from sage.all import *
>>> f = sage.categories.morphism.SetIsomorphism(Hom(ZZ, ZZ, Sets()),
...                                             operator.__neg__)
>>> f.is_surjective()
True
section()[source]#

Return a section of this morphism.

EXAMPLES:

sage: f = sage.categories.morphism.SetIsomorphism(Hom(ZZ, ZZ, Sets()),
....:                                             operator.__neg__)
sage: f._set_inverse(f)
sage: f.section() is f
True
>>> from sage.all import *
>>> f = sage.categories.morphism.SetIsomorphism(Hom(ZZ, ZZ, Sets()),
...                                             operator.__neg__)
>>> f._set_inverse(f)
>>> f.section() is f
True
class sage.categories.morphism.SetMorphism[source]#

Bases: Morphism

INPUT:

  • parent – a Homset

  • function – a Python function that takes elements of the domain as input and returns elements of the codomain.

EXAMPLES:

sage: from sage.categories.morphism import SetMorphism
sage: f = SetMorphism(Hom(QQ, ZZ, Sets()), numerator)
sage: f.parent()
Set of Morphisms from Rational Field to Integer Ring in Category of sets
sage: f.domain()
Rational Field
sage: f.codomain()
Integer Ring
sage: TestSuite(f).run()
>>> from sage.all import *
>>> from sage.categories.morphism import SetMorphism
>>> f = SetMorphism(Hom(QQ, ZZ, Sets()), numerator)
>>> f.parent()
Set of Morphisms from Rational Field to Integer Ring in Category of sets
>>> f.domain()
Rational Field
>>> f.codomain()
Integer Ring
>>> TestSuite(f).run()
sage.categories.morphism.is_Morphism(x)[source]#