higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
While every continuous map sends compact subsets to compact subsets, it is not true in general that the preimage of a compact set along a continuous map is compact.
A continuous function $f : X \to Y$ between topological spaces is proper if the inverse image of any compact subset is itself compact.
A notion of proper homotopy between proper maps leads to proper homotopy theory.
Similarly, one can consider the conditions on morphisms in other geometric situations, like algebraic geometry, and properness often either reflects or implies good behaviour with respect to the compact objects (cf. also proper push-forward).
A proper morphism of schemes is by definition a morphism $f:X\to Y$ which is
of finite type
universally closed (the latter means that for every $h: Z\to Y$ the pullback $h^*(f): Z\times_Y X\to Z$ is closed).
There is a classical and very practical valuative criterion of properness due Chevalley.
We say that a scheme $X$ is proper if the canonical map $X \to \operatorname{Spec} \mathbb{Z}$ to the terminal object is proper.
Proper schemes are analogous to compact topological spaces. This is one reason why one uses the terminology “quasi-compact” when referring to schemes whose underlying topological space is compact.
The base change formulas for cohomology for proper and for smooth morphisms of schemes motivated Grothendieck (in Pursuing Stacks) to define abstract proper and smooth functors in the setting of fibered categories; this is further expanded on in (Maltsiniotis).
(TO ADD: The definition of a proper dg algebra, proper dg category, proper A-inf cat ???)