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Madeleine Birchfield (Aug 26 2024 at 10:50):What is the origin of the [[sets cover]] axiom? Is it some theorem from classical algebraic topology that was later shown to not hold on other -toposes similar to the Whitehead theorem, or did it originate in dependent type theory i.e. when studying versions of the axiom of choice?
David Michael Roberts (Aug 26 2024 at 12:06):From memory it was from considering how in say Gpd, every object has a surjective-on-objects functor from a set. And then higher-categorical versions of these. So not so much a theorem in algebraic topology, but category theory.
@Mike Shulman I think might have a better idea about the history
Mike Shulman (Aug 26 2024 at 16:44):I think it did originate in HoTT, when trying to explain to people why proof techniques that work in classical algebraic topology, such as induction up dimensions of a CW complex, aren't going to work in HoTT.
Mike Shulman (Aug 26 2024 at 16:44):I wouldn't call it a "theorem" of classical algebraic topology so much as an obvious fact that no one even thought about stating explicitly.