# Functors¶

Functors

AUTHORS:

• David Kohel and William Stein
• David Joyner (2005-12-17): examples
• 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 loosing domain and codomain
sage.categories.functor.ForgetfulFunctor(domain, codomain)

Construct the forgetful function from one category to another.

INPUT:

C, D - two categories

OUTPUT:

A functor that returns the corresponding object of D for any element of C, by forgetting the extra structure.

ASSUMPTION:

The category C must be a sub-category of D.

EXAMPLES:

sage: rings = Rings()
sage: F = ForgetfulFunctor(rings, abgrps)
sage: 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


If both categories are equal, the forgetful functor is the same as the identity functor:

sage: ForgetfulFunctor(abgrps, abgrps) == IdentityFunctor(abgrps)
True

class sage.categories.functor.ForgetfulFunctor_generic

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 fields to Category of fields
sage: F(GF(3))
Finite Field of size 3

class sage.categories.functor.Functor

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 of self‘s domain, corresponding to x, or raise a TypeError.
• Default: Raise TypeError if x is not in self‘s domain.
• _apply_functor(self, x): Apply self to an object x of self‘s domain.
• Default: Conversion into self‘s codomain.
• _apply_functor_to_morphism(self, f): Apply self to a morphism f in self‘s domain. - Default: Return self(f.domain()).hom(f,self(f.codomain())).

EXAMPLES:

sage: rings  = Rings()
sage: F = ForgetfulFunctor(rings, abgrps)
sage: F.domain()
Category of rings
sage: F.codomain()
sage: from sage.categories.functor import is_Functor
sage: is_Functor(F)
True
sage: I = IdentityFunctor(abgrps)
sage: I
The identity functor on Category of commutative additive groups
sage: I.domain()
sage: is_Functor(I)
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


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)


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

codomain()

The codomain of self

EXAMPLE:

sage: F = ForgetfulFunctor(FiniteFields(),Fields())
sage: F.codomain()
Category of fields

domain()

The domain of self

EXAMPLE:

sage: F = ForgetfulFunctor(FiniteFields(),Fields())
sage: F.domain()
Category of finite fields

sage.categories.functor.IdentityFunctor(C)

Construct the identity functor of the given category.

INPUT:

A category, C.

OUTPUT:

The identity functor in C.

EXAPLES:

sage: rings = Rings()
sage: F = IdentityFunctor(rings)
sage: F(ZZ['x','y']) is ZZ['x','y']
True

class sage.categories.functor.IdentityFunctor_generic(C)

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


TESTS:

sage: R = IdentityFunctor(Rings())
sage: P, _ = QQ['t'].construction()
sage: R == P
False
sage: P == R
False
sage: R == QQ
False

sage.categories.functor.is_Functor(x)

Test whether the argument is a functor

NOTE:

There is a deprecation warning when using it from top level. Therefore we import it in our doc test.

EXAMPLES:

sage: from sage.categories.functor import is_Functor
sage: F1 = QQ.construction()[0]
sage: F1
FractionField
sage: is_Functor(F1)
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


Morphisms

#### Next topic

Coercion via Construction Functors