# Base class for maps¶

AUTHORS:

class sage.categories.map.FormalCompositeMap

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*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.)

EXAMPLES:

sage: f = QQ.coerce_map_from(ZZ)
sage: g = MatrixSpace(QQ, 2, 2).coerce_map_from(QQ)
sage: list((g*f).domains())
[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*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 can’t be determined.

EXAMPLES:

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()), phi1, phi2)
sage: c1.is_injective()
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)
sage: c2 = FormalCompositeMap(Hom(V1, V1, phi2.category_for()), phi2, psi1)
sage: c2.is_injective()
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([, ], QQ^1)
sage: c3 = FormalCompositeMap(Hom(V2, QQ^1, phi2.category_for()), psi2, psi1)
sage: c3.is_injective()
False

is_surjective()

Tell whether self is surjective.

It raises NotImplementedError if it can’t be determined.

EXAMPLES:

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


If both maps are surjective, the composition is surjective:

sage: phi32 = V3.hom(Matrix([[1, 2], [3, 4], [5, 6]]), V2)
sage: phi21 = V2.hom(Matrix([, ]), 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()), phi32, V2.hom(Matrix([, ]), 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()), V2.hom(Matrix([, ]), V1), V1.hom(Matrix([]), V1)).is_surjective()
False


Otherwise, surjectivity of the composition cannot be determined:

sage: FormalCompositeMap(Hom(V2, V1, phi32.category_for()),
....:     V2.hom(Matrix([[1, 1], [1, 1]]), V2),
....:     V2.hom(Matrix([, ]), 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*a^p.degree())
sage: g = S.hom([2*x])
sage: (f*g).then() == f
True

class sage.categories.map.Map

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)
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)
sage: phi.category_for()
Category of rings
sage: phi.category()
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.

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: 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)
sage: phi = CDF._internal_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


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
sage: _ = gc.collect()
sage: phi.parent()
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: 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 right 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)
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)
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

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: <type '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.