Towards congruences and quotients: - defined "regular categories" (no connection with "regular languages" though :-() (Definition 1.7.04): . finitely complete . coequalizers of kernel pairs exist; that means kernel pairs essentially give us congruence relations, and the coequalizers give us an "object of equivalence classes" in the category . regular epis are stable under pullbacks; this ensures that evert morhism has an "image-factorizations" through a minimal subobject of the codomain with a regular epi as first factor, since then also every extremal epi is regular; moreover, these factorizations are preserved by pulling back. - show that in regular categories every morphism has an essentially unique factorization as a regular epi followed by a mono. The proof takes the kerne pair of B --g-> C, and then forms the coequalizer B --e-> E of the parallel pair p_0 and p_1. The morphism E --m-> C induced by the universal property of the coequalizer and satisfying e;m = f is then shown to be mono. As a consequence, extremal epis in regular categories coincide with regular epis, so we have split epi ==> regular = strong = extremal epi ==> epi - consequence: congruences on X in regular categories bijectively correspond to regular=strong=extremal epis out of X, which in set-based categories means surjective homomorphisms. This works in mon and sgr. - But the Pin's definition of coongruences in mon resp. sgr looks different from what we have just seen. HW: show that both definitions of congruence coincide!