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).
Recall that a locale $L$ is given by a frame $O(L)$, its frame of opens, and that a continuous map $f$ from $K$ to $L$ is given by an adjunction $f^* \dashv f_* \colon O(K) \rightleftarrows O(L)$ such that the inverse image function $f^*$ preserves finitary meets (or equivalently is a frame homomorphism, since it must preserve all joins).
Such a map $f$ is proper iff the direct image function $f_*$ preserves directed joins (or equivalently is Scott-continuous, or equivalently is a morphism of preframes), and also satisfies the Frobenius reciprocity-like condition that $f_*(U\cup f^*(V)) = f_*(U) \cup V$ (which by itself states that the map is closed).
In particular, the map $L\to 1$ is proper iff $L$ is both compact and covert. But in this case the second condition is redundant, since every compact locale is automatically covert; see covert space for a proof.
Proper maps of locales can also be characterized as those that are universally closed, i.e. every pullback of them (along any map of locales) is closed.
Proper maps of locales can be generalized to geometric morphisms of Grothendieck toposes; see proper geometric morphism.
The topos-theoretic condition refers only to directed unions of subterminal objects, suggesting a stronger condition that it preserve all filtered colimits. This is a strictly stronger condition even for locales (i.e. localic toposes), called being tidy. In fact properness and tidiness are the first two rungs on an infinite ladder of higher properness for higher toposes.
(TO ADD: The definition of a proper dg algebra, proper dg category, proper A-inf cat ???)