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epimorphisms_are_surjective [2010/08/20 20:37] (current)
jipsen created
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 +=====Epimorphisms are surjective=====
 +
 +A morphism $h$ in a category is an \emph{epimorphism} if it is right-cancellative, i.e. for all
 +morphisms $f$, $g$ in the category $f\circ h=g\circ h$ implies $f=g$.
 +
 +A function $h:A\to B$ is \emph{surjective} (or \emph{onto}) if $B=f[A]=\{f(a): a\in A\}$,
 +i.e., for all $b\in B$ there exists $a\in A$ such that $f(a)=b$.
 +
 +\emph{Epimorphisms are surjective} in a (concrete) category of structures if the underlying function of every epimorphism is surjective.
 +
 +If a concrete category has the [[amalgamation property]] and all epimorphisms are surjective, then it has the [[strong amalgamation property]][(E. W. Kiss, L. Márki, P. Pröhle, W. Tholen, \emph{Categorical algebraic properties. A compendium on amalgamation, congruence extension, epimorphisms, residual smallness, and injectivity}, Studia Sci. Math. Hungar., \textbf{18}, 1982, 79-140 [[http://www.ams.org/mathscinet-getitem?mr=85k:18003|MRreview]])]