Kernel pair

Summary

The kernel pair of a morphism $f \colon X \to Y$ is the pullback of $f$ along itself. It records which pairs of elements $f$ sends to the same place. So it is the categorical form of the equivalence relation induced by a map. A morphism $f \colon X \to Y$ is a monomorphism exactly when $X$ with two identity maps is a kernel pair of $f$. In a regular category, a regular epimorphism is the coequalizer of its kernel pair.