The categorical dual of a monomorphism is an epimorphism, i . e . a monomorphism in a category " C " is an epimorphism in the dual category " C " op.
42.
Applying duality, this means that a morphism in some category " C " is a monomorphism if and only if the reverse morphism in the opposite category " C " op is an epimorphism.
43.
In the category of vector spaces over a field " K ", every monomorphism and every epimorphism splits; this follows from the fact that linear maps can be uniquely defined by specifying their values on a basis.
44.
This means that a ( homo ) morphism f : A \ to B is an epimorphism if, for any pair of morphisms from to any other object, the equality g \ circ f = h \ circ f implies.
45.
In particular, Mac Lane attempted to settle an ambiguity in usage for the terms epimorphism and monomorphism by introducing the terms " epic " and " monic, " but the distinction is not in common use.
46.
Such a factorization of an arbitrary morphism into an epimorphism followed by a monomorphism can be carried out in all abelian categories and also in all the concrete categories mentioned above in the Examples section ( though not in all concrete categories ).
47.
As an example, the inclusion of the rational numbers into the real numbers is a monomorphism and an epimorphism, but it is clearly not an isomorphism; this example shows that "'Met "'is not a balanced category.
48.
Here the hook arrow \ hookrightarrow indicates that the map 2?from "'Z "'to "'Z "'is a monomorphism, and the two-headed arrow \ twoheadrightarrow indicates an epimorphism ( the map mod 2 ).
49.
Note that it would better to use the notions of " epimorphism " and " monomorphism " because in general, you cannot define isomorphisms in this way unless you are dealing with a concrete category .-- T 17 : 31, 18 January 2009 ( UTC)
50.
In this book, Jordan introduced the notion of a simple group and epimorphism ( which he called " l'isomorphisme m�ri�drique " ), proved part of the Jordan H�lder theorem, and discussed matrix groups over finite fields as well as the Jordan normal form.
