21. For sets and vector spaces, every epimorphism is a split epimorphism , but this property is wrong for most common algebraic structures. 22. For projective resolutions, this condition is almost invisible : a projective pre-cover is simply an epimorphism from a projective module. 23. For example, whether or not a morphism of sheaves is a monomorphism, epimorphism , or isomorphism can be tested on the stalks. 24. Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories. 25. Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories. 26. This is an exact sequence because the image 2 "'Z "'of the monomorphism is the kernel of the epimorphism . 27. Conversely an epimorphism is called " normal " ( or " conormal " ) if it is the cokernel of some morphism. 28. A category is called " conormal " if every epimorphism is normal ( e . g . the category of groups is conormal ). 29. The most basic example of an epimorphism ( category theory meaning ), which is not surjective, is the "'Q " '. 30. For a morphism f \ colon B \ to A, this is precisely what it means for " f " to be an epimorphism .
