Base class for maps

AUTHORS:

• Robert Bradshaw: initial implementation

• Sebastien Besnier (2014-05-5): FormalCompositeMap contains a list of Map instead of only two Map. See github issue #16291.

• Sebastian Oehms (2019-01-19): section() added to FormalCompositeMap. See github issue #27081.

class sage.categories.map.FormalCompositeMap

Bases: Map

Formal composite maps.

A formal composite map is formed by two maps, so that the codomain of the first map is contained in the domain of the second map.

Note

When calling a composite with additional arguments, these arguments are only passed to the second underlying map.

EXAMPLES:

sage: R.<x> = QQ[]
sage: S.<a> = QQ[]
sage: from sage.categories.morphism import SetMorphism
sage: f = SetMorphism(Hom(R, S, Rings()), lambda p: p[0]*a^p.degree())
sage: g = S.hom([2*x])
sage: f*g
Composite map:
  From: Univariate Polynomial Ring in a over Rational Field
  To:   Univariate Polynomial Ring in a over Rational Field
  Defn:   Ring morphism:
          From: Univariate Polynomial Ring in a over Rational Field
          To:   Univariate Polynomial Ring in x over Rational Field
          Defn: a |--> 2*x
        then
          Generic morphism:
          From: Univariate Polynomial Ring in x over Rational Field
          To:   Univariate Polynomial Ring in a over Rational Field
sage: g*f
Composite map:
  From: Univariate Polynomial Ring in x over Rational Field
  To:   Univariate Polynomial Ring in x over Rational Field
  Defn:   Generic morphism:
          From: Univariate Polynomial Ring in x over Rational Field
          To:   Univariate Polynomial Ring in a over Rational Field
        then
          Ring morphism:
          From: Univariate Polynomial Ring in a over Rational Field
          To:   Univariate Polynomial Ring in x over Rational Field
          Defn: a |--> 2*x
sage: (f*g)(2*a^2+5)
5*a^2
sage: (g*f)(2*x^2+5)
20*x^2

domains()

Iterate over the domains of the factors of this map.

(This is useful in particular to check for loops in coercion maps.)

See also

Map.domains()

EXAMPLES:

sage: f = QQ.coerce_map_from(ZZ)
sage: g = MatrixSpace(QQ, 2, 2).coerce_map_from(QQ)                         # needs sage.modules
sage: list((g * f).domains())                                               # needs sage.modules
[Integer Ring, Rational Field]

first()

Return the first map in the formal composition.

If self represents \(f_n \circ f_{n-1} \circ \cdots \circ f_1 \circ f_0\), then self.first() returns \(f_0\). We have self == self.then() * self.first().

EXAMPLES:

sage: R.<x> = QQ[]
sage: S.<a> = QQ[]
sage: from sage.categories.morphism import SetMorphism
sage: f = SetMorphism(Hom(R, S, Rings()), lambda p: p[0]*a^p.degree())
sage: g = S.hom([2*x])
sage: fg = f * g
sage: fg.first() == g
True
sage: fg == fg.then() * fg.first()
True

is_injective()

Tell whether self is injective.

It raises NotImplementedError if it cannot be determined.

EXAMPLES:

sage: # needs sage.modules
sage: V1 = QQ^2
sage: V2 = QQ^3
sage: phi1 = (QQ^1).hom(Matrix([[1, 1]]), V1)
sage: phi2 = V1.hom(Matrix([[1, 2, 3], [4, 5, 6]]), V2)

If both constituents are injective, the composition is injective:

sage: from sage.categories.map import FormalCompositeMap
sage: c1 = FormalCompositeMap(Hom(QQ^1, V2, phi1.category_for()),           # needs sage.modules
....:                         phi1, phi2)
sage: c1.is_injective()                                                     # needs sage.modules
True

If it cannot be determined whether the composition is injective, an error is raised:

sage: psi1 = V2.hom(Matrix([[1, 2], [3, 4], [5, 6]]), V1)                   # needs sage.modules
sage: c2 = FormalCompositeMap(Hom(V1, V1, phi2.category_for()),             # needs sage.modules
....:                         phi2, psi1)
sage: c2.is_injective()                                                     # needs sage.modules
Traceback (most recent call last):
...
NotImplementedError: not enough information to deduce injectivity

If the first map is surjective and the second map is not injective, then the composition is not injective:

sage: psi2 = V1.hom([[1], [1]], QQ^1)                                       # needs sage.modules
sage: c3 = FormalCompositeMap(Hom(V2, QQ^1, phi2.category_for()),           # needs sage.modules
....:                         psi2, psi1)
sage: c3.is_injective()                                                     # needs sage.modules
False

is_surjective()

Tell whether self is surjective.

It raises NotImplementedError if it cannot be determined.

EXAMPLES:

sage: from sage.categories.map import FormalCompositeMap
sage: V3 = QQ^3                                                             # needs sage.modules
sage: V2 = QQ^2                                                             # needs sage.modules
sage: V1 = QQ^1                                                             # needs sage.modules

If both maps are surjective, the composition is surjective:

sage: # needs sage.modules
sage: phi32 = V3.hom(Matrix([[1, 2], [3, 4], [5, 6]]), V2)
sage: phi21 = V2.hom(Matrix([[1], [1]]), V1)
sage: c_phi = FormalCompositeMap(Hom(V3, V1, phi32.category_for()),
....:                            phi32, phi21)
sage: c_phi.is_surjective()
True

If the second map is not surjective, the composition is not surjective:

sage: FormalCompositeMap(Hom(V3, V1, phi32.category_for()),                 # needs sage.modules
....:                    phi32,
....:                    V2.hom(Matrix([[0], [0]]), V1)).is_surjective()
False

If the second map is an isomorphism and the first map is not surjective, then the composition is not surjective:

sage: FormalCompositeMap(Hom(V2, V1, phi32.category_for()),                 # needs sage.modules
....:                    V2.hom(Matrix([[0], [0]]), V1),
....:                    V1.hom(Matrix([[1]]), V1)).is_surjective()
False

Otherwise, surjectivity of the composition cannot be determined:

sage: FormalCompositeMap(Hom(V2, V1, phi32.category_for()),                 # needs sage.modules
....:     V2.hom(Matrix([[1, 1], [1, 1]]), V2),
....:     V2.hom(Matrix([[1], [1]]), V1)).is_surjective()
Traceback (most recent call last):
...
NotImplementedError: not enough information to deduce surjectivity

section()

Compute a section map from sections of the factors of self if they have been implemented.

EXAMPLES:

sage: P.<x> = QQ[]
sage: incl = P.coerce_map_from(ZZ)
sage: sect = incl.section(); sect
Composite map:
  From: Univariate Polynomial Ring in x over Rational Field
  To:   Integer Ring
  Defn:   Generic map:
          From: Univariate Polynomial Ring in x over Rational Field
          To:   Rational Field
        then
          Generic map:
          From: Rational Field
          To:   Integer Ring
sage: p = x + 5; q = x + 2
sage: sect(p-q)
3

the following example has been attached to _integer_() of sage.rings.polynomial.polynomial_element.Polynomial before (see comment there):

sage: k = GF(47)
sage: R.<x> = PolynomialRing(k)
sage: R.coerce_map_from(ZZ).section()
Composite map:
  From: Univariate Polynomial Ring in x over Finite Field of size 47
  To:   Integer Ring
  Defn:   Generic map:
          From: Univariate Polynomial Ring in x over Finite Field of size 47
          To:   Finite Field of size 47
        then
          Lifting map:
          From: Finite Field of size 47
          To:   Integer Ring
sage: ZZ(R(45))                 # indirect doctest
45
sage: ZZ(3*x + 45)              # indirect doctest
Traceback (most recent call last):
...
TypeError: not a constant polynomial

then()

Return the tail of the list of maps.

If self represents \(f_n \circ f_{n-1} \circ \cdots \circ f_1 \circ f_0\), then self.first() returns \(f_n \circ f_{n-1} \circ \cdots \circ f_1\). We have self == self.then() * self.first().

EXAMPLES:

sage: R.<x> = QQ[]
sage: S.<a> = QQ[]
sage: from sage.categories.morphism import SetMorphism
sage: f = SetMorphism(Hom(R, S, Rings()), lambda p: p[0]*a^p.degree())
sage: g = S.hom([2*x])
sage: (f*g).then() == f
True

sage: f = QQ.coerce_map_from(ZZ)
sage: f = f.extend_domain(ZZ).extend_codomain(QQ)
sage: f.then()
Composite map:
From: Integer Ring
To:   Rational Field
Defn:   Natural morphism:
From: Integer Ring
To:   Rational Field
then
Identity endomorphism of Rational Field

class sage.categories.map.Map

Bases: Element

Basic class for all maps.

Note

The call method is of course not implemented in this base class. This must be done in the sub classes, by overloading _call_ and possibly also _call_with_args.

EXAMPLES:

Usually, instances of this class will not be constructed directly, but for example like this:

sage: from sage.categories.morphism import SetMorphism
sage: X.<x> = ZZ[]
sage: Y = ZZ
sage: phi = SetMorphism(Hom(X, Y, Rings()), lambda p: p[0])
sage: phi(x^2+2*x-1)
-1
sage: R.<x,y> = QQ[]
sage: f = R.hom([x+y, x-y], R)
sage: f(x^2+2*x-1)
x^2 + 2*x*y + y^2 + 2*x + 2*y - 1

category_for()

Returns the category self is a morphism for.

Note

This is different from the category of maps to which this map belongs as an object.

EXAMPLES:

sage: from sage.categories.morphism import SetMorphism
sage: X.<x> = ZZ[]
sage: Y = ZZ
sage: phi = SetMorphism(Hom(X, Y, Rings()), lambda p: p[0])
sage: phi.category_for()
Category of rings
sage: phi.category()
Category of homsets of unital magmas and additive unital additive magmas
sage: R.<x,y> = QQ[]
sage: f = R.hom([x+y, x-y], R)
sage: f.category_for()
Join of Category of unique factorization domains
and Category of commutative algebras
over (number fields and quotient fields and metric spaces)
and Category of infinite sets
sage: f.category()
Category of endsets of unital magmas
 and right modules over (number fields and quotient fields and metric spaces)
 and left modules over (number fields and quotient fields and metric spaces)

FIXME: find a better name for this method

codomain

domain

domains()

Iterate over the domains of the factors of a (composite) map.

This default implementation simply yields the domain of this map.

See also

FormalCompositeMap.domains()

EXAMPLES:

sage: list(QQ.coerce_map_from(ZZ).domains())
[Integer Ring]

extend_codomain(new_codomain)

INPUT:

• self – a member of Hom(X, Y)

• new_codomain – an object Z such that there is a canonical coercion \(\phi\) in Hom(Y, Z)

OUTPUT:

An element of Hom(X, Z) obtained by composing self with \(\phi\). If no canonical \(\phi\) exists, a TypeError is raised.

EXAMPLES:

sage: mor = QQ.coerce_map_from(ZZ)
sage: mor.extend_codomain(RDF)
Composite map:
  From: Integer Ring
  To:   Real Double Field
  Defn:   Natural morphism:
          From: Integer Ring
          To:   Rational Field
        then
          Native morphism:
          From: Rational Field
          To:   Real Double Field
sage: mor.extend_codomain(GF(7))
Traceback (most recent call last):
...
TypeError: No coercion from Rational Field to Finite Field of size 7

extend_domain(new_domain)

INPUT:

• self – a member of Hom(Y, Z)

• new_codomain – an object X such that there is a canonical coercion \(\phi\) in Hom(X, Y)

OUTPUT:

An element of Hom(X, Z) obtained by composing self with \(\phi\). If no canonical \(\phi\) exists, a TypeError is raised.

EXAMPLES:

sage: # needs sage.rings.complex_double
sage: mor = CDF.coerce_map_from(RDF)
sage: mor.extend_domain(QQ)
Composite map:
  From: Rational Field
  To:   Complex Double Field
  Defn:   Native morphism:
          From: Rational Field
          To:   Real Double Field
        then
          Native morphism:
          From: Real Double Field
          To:   Complex Double Field
sage: mor.extend_domain(ZZ['x'])
Traceback (most recent call last):
...
TypeError: No coercion from Univariate Polynomial Ring in x over Integer Ring
to Real Double Field

is_surjective()

Tells whether the map is surjective (not implemented in the base class).

parent()

Return the homset containing this map.

Note

The method _make_weak_references(), that is used for the maps found by the coercion system, needs to remove the usual strong reference from the coercion map to the homset containing it. As long as the user keeps strong references to domain and codomain of the map, we will be able to reconstruct the homset. However, a strong reference to the coercion map does not prevent the domain from garbage collection!

EXAMPLES:

sage: Q = QuadraticField(-5)                                                # needs sage.rings.number_field
sage: phi = CDF._internal_convert_map_from(Q)                               # needs sage.rings.number_field
sage: print(phi.parent())                                                   # needs sage.rings.number_field
Set of field embeddings
 from Number Field in a with defining polynomial x^2 + 5
      with a = 2.236067977499790?*I
   to Complex Double Field

We now demonstrate that the reference to the coercion map \(\phi\) does not prevent \(Q\) from being garbage collected:

sage: import gc
sage: del Q                                                                 # needs sage.rings.number_field
sage: _ = gc.collect()
sage: phi.parent()                                                          # needs sage.rings.number_field
Traceback (most recent call last):
...
ValueError: This map is in an invalid state,
the domain has been garbage collected

You can still obtain copies of the maps used by the coercion system with strong references:

sage: # needs sage.rings.number_field
sage: Q = QuadraticField(-5)
sage: phi = CDF.convert_map_from(Q)
sage: print(phi.parent())
Set of field embeddings
 from Number Field in a with defining polynomial x^2 + 5
      with a = 2.236067977499790?*I
   to Complex Double Field
sage: import gc
sage: del Q
sage: _ = gc.collect()
sage: phi.parent()
Set of field embeddings
 from Number Field in a with defining polynomial x^2 + 5
      with a = 2.236067977499790?*I
   to Complex Double Field

post_compose(left)

INPUT:

• self – a Map in some Hom(X, Y, category_right)

• left – a Map in some Hom(Y, Z, category_left)

Returns the composition of self followed by left as a morphism in Hom(X, Z, category) where category is the meet of category_left and category_right.

Caveat: see the current restrictions on Category.meet()

EXAMPLES:

sage: from sage.categories.morphism import SetMorphism
sage: X.<x> = ZZ[]
sage: Y = ZZ
sage: Z = QQ
sage: phi_xy = SetMorphism(Hom(X, Y, Rings()), lambda p: p[0])
sage: phi_yz = SetMorphism(Hom(Y, Z, Monoids()), lambda y: QQ(y**2))
sage: phi_xz = phi_xy.post_compose(phi_yz); phi_xz
Composite map:
  From: Univariate Polynomial Ring in x over Integer Ring
  To:   Rational Field
  Defn:   Generic morphism:
          From: Univariate Polynomial Ring in x over Integer Ring
          To:   Integer Ring
        then
          Generic morphism:
          From: Integer Ring
          To:   Rational Field
sage: phi_xz.category_for()
Category of monoids

pre_compose(right)

INPUT:

• self – a Map in some Hom(Y, Z, category_left)

• left – a Map in some Hom(X, Y, category_right)

Returns the composition of right followed by self as a morphism in Hom(X, Z, category) where category is the meet of category_left and category_right.

EXAMPLES:

sage: from sage.categories.morphism import SetMorphism
sage: X.<x> = ZZ[]
sage: Y = ZZ
sage: Z = QQ
sage: phi_xy = SetMorphism(Hom(X, Y, Rings()), lambda p: p[0])
sage: phi_yz = SetMorphism(Hom(Y, Z, Monoids()), lambda y: QQ(y**2))
sage: phi_xz = phi_yz.pre_compose(phi_xy); phi_xz
Composite map:
  From: Univariate Polynomial Ring in x over Integer Ring
  To:   Rational Field
  Defn:   Generic morphism:
          From: Univariate Polynomial Ring in x over Integer Ring
          To:   Integer Ring
        then
          Generic morphism:
          From: Integer Ring
          To:   Rational Field
sage: phi_xz.category_for()
Category of monoids

section()

Return a section of self.

Note

By default, it returns None. You may override it in subclasses.

class sage.categories.map.Section

Bases: Map

A formal section of a map.

Note

Call methods are not implemented for the base class Section.

EXAMPLES:

sage: from sage.categories.map import Section
sage: R.<x,y> = ZZ[]
sage: S.<a,b> = QQ[]
sage: f = R.hom([a+b, a-b])
sage: sf = Section(f); sf
Section map:
  From: Multivariate Polynomial Ring in a, b over Rational Field
  To:   Multivariate Polynomial Ring in x, y over Integer Ring
sage: sf(a)
Traceback (most recent call last):
...
NotImplementedError: <class 'sage.categories.map.Section'>

inverse()

Return inverse of self.

sage.categories.map.is_Map(x)

Auxiliary function: Is the argument a map?

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: f = R.hom([x+y, x-y], R)
sage: from sage.categories.map import is_Map
sage: is_Map(f)
True

sage.categories.map.unpickle_map(_class, parent, _dict, _slots)

Auxiliary function for unpickling a map.