I don't have very good intuition for split epis/monos
suppose i have a fiber bundle E->B
a splitting of this is a global section, so it's a split epi iff it *admits a global section*. This seems like... kind of a weird thing to be *such* a fundamental categorical concept
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i guess i have two intuitions here; one is that split epis/monos are equational and so preserved by arbitrary functors (which characterizes them), and the other is about splitting idempotents
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split epis and monos are sort of the closest you can get 1-categorically to adjoint functors. the two stories are surprisingly parallel. the idempotent corresponds to the monad / comonad and so forth
