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19.2 Homsets

Module: sage.categories.homset

Homsets

Author Log:

Module-level Functions

End( X, [cat=None])
Create the set of endomorphisms of X in the category cat. 

INPUT:
    X -- anything
    cat -- (optional) category in which to coerce X 

OUTPUT:
    a set of endomorphisms in cat


sage: V = VectorSpace(QQ, 3)
sage: End(V)
Set of Morphisms from Vector space of dimension 3 over Rational
Field to Vector space of dimension 3 over Rational Field in
Category of vector spaces over Rational Field


sage: G = SymmetricGroup(3)
sage: S = End(G); S
Set of Morphisms from SymmetricGroup(3) to SymmetricGroup(3) in Category of
groups
sage: is_Endset(S)
True
sage: S.domain()
Symmetric group of order 3! as a permutation group

Homsets are not objects in their category. They are currently sets. 

sage: S.category()
Category of sets
sage: S.domain().category()
Category of groups

Hom( X, Y, [cat=None])
Create the space of homomorphisms from X to Y in the category cat. 

INPUT:
    X -- anything
    Y -- anything
    cat -- (optional) category in which the morphisms must be

OUTPUT:
    a homset in cat


sage: V = VectorSpace(QQ,3)
sage: Hom(V, V)
Set of Morphisms from Vector space of dimension 3 over Rational
Field to Vector space of dimension 3 over Rational Field in
Category of vector spaces over Rational Field
sage: G = SymmetricGroup(3)
sage: Hom(G, G)
Set of Morphisms from SymmetricGroup(3) to SymmetricGroup(3) in Category of
groups
sage: Hom(ZZ, QQ, Sets())
Set of Morphisms from Integer Ring to Rational Field in Category of sets

end( X, f)
Return End(X)(f), where f is data that defines an element of End(X). 

sage: R, x = PolynomialRing(QQ,'x').objgen()
sage: phi = end(R, [x + 1])
sage: phi
Ring endomorphism of Univariate Polynomial Ring in x over Rational Field
  Defn: x |--> x + 1
sage: phi(x^2 + 5)
x^2 + 2*x + 6

hom( X, Y, f)
Return Hom(X,Y)(f), where f is data that defines an element of Hom(X,Y). 

sage: R, x = PolynomialRing(QQ,'x').objgen()
sage: phi = hom(R, QQ, [2])
sage: phi(x^2 + 3)
7

is_Endset( x)
Return True if x is a set of endomorphisms in a category.

is_Homset( x)
Return True if x is a set of homomorphisms in a category.

Class: Homset

class Homset
The class for collections of morphisms in a category. 

sage: H = Hom(QQ^2, QQ^3)
sage: loads(H.dumps()) == H
True
sage: E = End(AffineSpace(2, names='x,y'))
sage: loads(E.dumps()) == E
True
Homset( self, X, Y, [cat=None], [check=True])

Functions: codomain,$ $ domain,$ $ identity,$ $ is_endomorphism_set,$ $ natural_map,$ $ reversed

is_endomorphism_set( self)
Return True if the domain and codomain of self are the same object.

reversed( self)
Return the corresponding homset, but with the domain and codomain reversed.

Special Functions: __call__,$ $ __cmp__,$ $ __contains__,$ $ _repr_

__call__( self, x, [y=None], [check=True])
Construct a morphism in this homset from x if possible. 

sage: H = Hom(SymmetricGroup(4), SymmetricGroup(7))
sage: phi = Hom(SymmetricGroup(5), SymmetricGroup(6)).natural_map()
sage: phi
Coercion morphism:
  From: SymmetricGroup(5)
  To:   SymmetricGroup(6)
sage: H(phi)
Composite morphism:
  From: SymmetricGroup(4)
  To:   SymmetricGroup(7)
  Defn:   Composite morphism:
          From: SymmetricGroup(4)
          To:   SymmetricGroup(6)
          Defn:   Coercion morphism:
                  From: SymmetricGroup(4)
                  To:   SymmetricGroup(5)
                then
                  Coercion morphism:
                  From: SymmetricGroup(5)
                  To:   SymmetricGroup(6)
        then
          Coercion morphism:
          From: SymmetricGroup(6)
          To:   SymmetricGroup(7)

Author: Robert Bradshaw

Class: HomsetWithBase

class HomsetWithBase
HomsetWithBase( self, X, Y, [cat=None], [check=True], [base=None])

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