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For questions about or involving fibrations which are maps which satisfy the homotopy lifting property for all spaces.
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Can Homotopy Type Theory or algebraic geometry deal with homotopy fibers in terms of families?
to enlarge your notion of equivalence, for example you could state that all fibers of the universal family are equivalent, e.g. by saying that an equivalence of $E$ and $E^\prime$ can be given as any fibration …