Epimorphism

Remarks

Equivalently, for every object $Y$ the map $\Hom_{\mathcal{C}}(B, Y) \to \Hom_{\mathcal{C}}(A, Y)$, $g \mapsto g \circ f$, is injective. One writes $f \colon A \twoheadrightarrow B$, or says that $f$ is epic. Example: in the category of sets, every surjective function $f$ is an epimorphism. If $g_1 \circ f = g_2 \circ f$ and $b \in B$, write $b = f(a)$; then $g_1(b) = g_1(f(a)) = g_2(f(a)) = g_2(b)$. Non-example: the inclusion $f \colon \set{0} \to \set{0, 1}$ is not an epimorphism of sets. The identity of $\set{0, 1}$ and the constant map with value $0$ agree after $f$ but are different. In the category of rings with unit, the inclusion $\Z \to \Q$ is an epimorphism that is not surjective. Two ring homomorphisms $g_1, g_2 \colon \Q \to R$ that agree on $\Z$ agree on $m/n$, since $g_i(m/n) = g_i(m) g_i(n)^{-1}$.