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

AUTHORS:

William Stein (2005): initial version
David Joyner (2005-12-17): added examples
Robert Bradshaw (2007-06-25): Pyrexification

class sage.categories.morphism.CallMorphism#: Bases: sage.categories.morphism.Morphism

class sage.categories.morphism.FormalCoercionMorphism#: Bases: sage.categories.morphism.Morphism

class sage.categories.morphism.IdentityMorphism#

Bases: sage.categories.morphism.Morphism

is_identity()#

Return True if this morphism is the identity morphism.

EXAMPLES:

sage: E = End(Partitions(5))
sage: E.identity().is_identity()
True

Check that trac ticket #15478 is fixed:

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

is_injective()#

Return whether this morphism is injective.

EXAMPLES:

sage: Hom(ZZ, ZZ).identity().is_injective()
True

is_surjective()#

Return whether this morphism is surjective.

EXAMPLES:

sage: Hom(ZZ, ZZ).identity().is_surjective()
True

section()#

Return a section of this morphism.

EXAMPLES:

sage: T = Hom(ZZ, ZZ).identity()
sage: T.section() is T
True

class sage.categories.morphism.Morphism#

Bases: sage.categories.map.Map

category()#

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
 (euclidean domains and infinite enumerated sets and metric spaces)
 and left modules over (euclidean domains
 and infinite enumerated sets and metric spaces)

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

is_endomorphism()#

Return True if this morphism is an endomorphism.

EXAMPLES:

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

sage: K = CyclotomicField(12)
sage: L = CyclotomicField(132)
sage: phi = L._internal_coerce_map_from(K)
sage: phi.is_endomorphism()
False

is_identity()#

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

A morphism between two different spaces cannot be the identity:

sage: R2.<t2> = QQ[]
sage: h = R.hom([t2])
sage: h.is_identity()
False

pushforward(I)#

register_as_coercion()#

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'

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()

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

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

register_as_conversion()#

(see sage.structure.coerce).

EXAMPLES:

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

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

class sage.categories.morphism.SetMorphism#

Bases: sage.categories.morphism.Morphism

INPUT:

parent – a Homset
function – a Python function that takes elements of the domain as input and returns elements of the domain.

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()

sage.categories.morphism.is_Morphism(x)#