On page 367 of Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory there is mentioned a kernel co-kernel correpsondence, which says there's an equivalence between normal monos and epis, and furthermore that effective equivalence relations are equivalent to regular epis. What exactly are the categories involved, and how, for instance, are the kernel and cokernel a category equivalence?
2026-03-27 20:01:03.1774641663
Kernel cokernel correspondence?
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The categories involved are the full subcategories of the morphism category of $C$. Its objects are morphisms of $C$ and its arrows are commutative squares.
As for the correspondence, just think of kernel of $f$ not as of an object but as of a morphism ("the inclusion of kernel into the domain of $f$") and likewise think of cokernel. Then a kernel of a normal epi is a normal mono and a cokernel of normal mono is a normal epi, and taking (co)kernels is functorial.