Epimorphism

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In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism f : XY which is right-cancellative in the sense that, for all morphisms g1, g2 : YZ,

Epimorphisms are analogues of surjective functions, but they are not exactly the same. The dual of an epimorphism is a monomorphism (i.e. an epimorphism in a category C is a monomorphism in the dual category Cop).

Many authors in abstract algebra and universal algebra define an epimorphism simply as an onto or surjective homomorphism. Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories. In this article, the term "epimorphism" will be used in the sense of category theory given above. For more on this, see the section on Terminology below.

Contents

Examples

Every morphism in a concrete category whose underlying function is surjective is an epimorphism. In many concrete categories of interest the converse is also true. For example, in the following categories, the epimorphisms are exactly those morphisms which are surjective on the underlying sets:

  • Set, sets and functions. To prove that every epimorphism f: XY in Set is surjective, we compose it with both the characteristic function g1: Y → {0,1} of the image f(X) and the map g2: Y → {0,1} that is constant 1.
  • Rel, sets with binary relations and relation preserving functions. Here we can use the same proof as for Set, equipping {0,1} with the full relation {0,1}×{0,1}.
  • Pos, partially ordered sets and monotone functions. If f : (X,≤) → (Y,≤) is not surjective, pick y0 in Y \ f(X) and let g1 : Y → {0,1} be the characteristic function of {y | y0y} and g2 : Y → {0,1} the characteristic function of {y | y0 < y}. These maps are monotone if {0,1} is given the standard ordering 0 < 1.
  • Grp, groups and group homomorphisms. The result that every epimorphism in Grp is surjective is due to Otto Schreier (he actually proved more, showing that every subgroup is an equalizer using the free product with one amalgamated subgroup); an elementary proof can be found in (Linderholm 1970).
  • FinGrp, finite groups and group homomorphisms. Also due to Schreier; the proof given in (Linderholm 1970) establishes this case as well.
  • Ab, abelian groups and group homomorphisms.
  • K-Vect, vector spaces over a field K and K-linear transformations.
  • Mod-R, right modules over a ring R and module homomorphisms. This generalizes the two previous examples; to prove that every epimorphism f: XY in Mod-R is surjective, we compose it with both the canonical quotient map g 1: YY/f(X) and the zero map g2: YY/f(X).
  • Top, topological spaces and continuous functions. To prove that every epimorphism in Top is surjective, we proceed exactly as in Set, giving {0,1} the indiscrete topology which ensures that all considered maps are continuous.
  • HComp, compact Hausdorff spaces and continuous functions. If f: XY is not surjective, let y in Y-fX. Since fX is closed, by Urysohn's Lemma there is a continuous function g1:Y → [0,1] such that g1 is 0 on fX and 1 on y. We compose f with both g1 and the zero function g2: Y → [0,1].

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