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# Functors

Functors

AUTHORS:

- David Kohel and William Stein
- David Joyner (2005-12-17): examples
- Robert Bradshaw (2007-06-23): Pyrexify
- 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: abgrps = CommutativeAdditiveGroups()
      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`

  Bases: `sage.categories.functor.Functor`

  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`

  Bases: `sage.structure.sage_object.SageObject`

  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: abgrps = CommutativeAdditiveGroups()
      sage: F = ForgetfulFunctor(rings, abgrps)
      sage: F.domain()
      Category of rings
      sage: F.codomain()
      Category of commutative additive groups
      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()
      Category of commutative additive groups
      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)

  Bases: `sage.categories.functor.ForgetfulFunctor_generic`

  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

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