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Concept
Epimorphism
Summary
In , an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism f : X → Y that is right-cancellative in the sense that, for all objects Z and all morphisms ,
Epimorphisms are categorical analogues of onto or surjective functions (and in the the concept corresponds exactly to the surjective functions), but they may not exactly coincide in all contexts; for example, the inclusion is a ring epimorphism. The of an epimorphism is a monomorphism (i.e. an epimorphism in a C is a monomorphism in the 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 below.
Every morphism in a 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 that are surjective on the underlying sets:
sets and functions. To prove that every epimorphism f: X → Y 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 | y0 ≤ y} 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.
groups and group homomorphisms.
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