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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as opposed to say "admits enough global sections that it's forced to be a trivial bundle"
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now i have a short exact sequence of abelian gronps (or whatever), which i tend to think of as something like a fiber bundle (it will literally give a fibration of eilenberg-maclane spectra i think), a splitting of *that* forces it to be a trivial bundle (= direct product)
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what's special about abelian groups that makes this happen?
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abelian groups are enriched over abelian groups, which means the idempotent m corresponding to a split epi or a split mono has a "complementary" idempotent 1 - m which exhibits a direct sum