A morphism $f \colon A \to B$ in a category is called a monic morphism, or 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 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 sections and 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.