"Proper" is an adjective used to describe a morphism of spaces—topological spaces, schemes, locales, etc—that is sufficiently nice and has some neat properties. Between topological spaces a morphism is proper if the preimage of compact set is compact, which is a clean definition, but it doesn't give me any geometric intuition that generalizes to schemes or beyond. Similarly the definitions I've seen for a proper morphisms of schemes hasn't helped improve my intuition.

What are some flags/heuristics/features of a morphism that I should look out for to decide whether or not it's proper? What are some illustrative examples or non-examples of proper morphisms, in whatever setting, that could help build my geometric intuition here? This answer over on MathOverflow helps a bit, but only a bit,

  • $\begingroup$ I suspect that "proper" is a term that has just been indroduced independently in different contexts with no proper consideration for the bigger picture. $\endgroup$ – Arthur Jan 9 at 15:00
  • 4
    $\begingroup$ @Arthur this nLab page and the other answer to that MO question suggest otherwise. $\endgroup$ – Mike Pierce Jan 9 at 15:01

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