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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vaguely related question: if i look at groups instead of abelian ones, then a split exact sequence mean u have a semi-direct product
i wanna figure out what this means in terms of bundles of deloopings or smth
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it means the following: a SES of groups 1 -> N -> G -> H -> 1 corresponds to a fiber sequence BN -> BG -> BH classified by a homotopy action of H on BN, and the SES splits ifff this homotopy action has a homotopy fixed point iff the action strictifies to an action on N
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morally this just comes from the observation that taking (derived) global sections of a (locally constant oo-)sheaf on BN corresponds to taking (derived) fixed points
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oh, and i forgot to fill in a step: and if H has a homotopy fixed point acting on BN then it acts on pi_1(BN, the fixed point) = N on the nose
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lol there are three f's in my iffffff