Monomorphism

Remarks

Equivalently, for every object $X$ the map $\Hom_{\mathcal{C}}(X, A) \to \Hom_{\mathcal{C}}(X, B)$, $g \mapsto f \circ g$, is injective. One writes $f \colon A \rightarrowtail B$, or says that $f$ is monic. Example: in the category of sets, every injective function $f$ is a monomorphism. If $f \circ g_1 = f \circ g_2$, then $f(g_1(x)) = f(g_2(x))$ for each $x \in X$, so $g_1(x) = g_2(x)$ by injectivity. Non-example: the map $f \colon \set{1, 2} \to \set{0}$ is not a monomorphism. The two maps $g_1, g_2 \colon \set{0} \to \set{1, 2}$ with $g_1(0) = 1$ and $g_2(0) = 2$ satisfy $f \circ g_1 = f \circ g_2$ but $g_1 \ne g_2$. Every isomorphism is a monomorphism, and a composite of monomorphisms is a monomorphism. If $g \circ f$ is a monomorphism, then $f$ is a monomorphism.