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!