monic Monic 2002-02-10 15:48:24 2008-09-15 17:09:59 Definition split monomorphism section coretraction % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A morphism $f \colon A \to B$ in a category is called a {\em monic} morphism, or \emph{monomorphism}, if it can be cancelled from the left --- for any object $C$ and any morphisms $g_1, g_2 \colon\ C\to A$ we have $f \circ g_1 = f \circ g_2$ if and only if $g_1 = g_2$. A morphism $f : A \to B$ in a category is called a \emph{split monomorphism} if there exists a morphism $g \colon B \to A$ such that $g \circ f = \operatorname{id}_A$. Note that every split monomorphism is a monomorphism; if $f$ is a split monomorphism and $f \circ h = f \circ k$, then one has $g \circ (f \circ h) = g \circ (f \circ k)$. By associativity, $(g \circ f) \circ h = (g \circ f) \circ k$; by definition of split monomorphism, $\operatorname{id}_a \circ h = \operatorname{id}_a \circ k$; by definition of identity, $h = k$, so $f$ is a monomorphism. Split monomorphisms are also known as \emph{sections} and \emph{coretractions}. The notion of epimorphism is dual to that of monomorphism. An epimorphism of a category is a monomorphism of the dual category and vice versa. A monomorphism in the category of sets is simply a one-to-one function. Moreover, in the category of sets all monomorphisms are split monomorphisms.