📝Category Theory: Monomorphism

\begin{tikzcd} Z \arrow[r, "g_1", shift left] \arrow[r, "g_2", shift right, swap] & X \arrow[r, "f"] & Y \end{tikzcd}

f:XYf : X \to Y is a monomorphism iff (g1,g2:ZX)  fg1=fg2    g1=g2\forall(g_1, g_2 : Z \to X)\; f \circ g_1 = f \circ g_2 \implies g_1 = g_2

(The rule to remember is that you can cancel ff from the left side of equation.)

Monomorphism is defined for all categories (but might be absent), and is a generalization of injective functions from Set Theory (Set theory: functions).

See also: