derived smooth geometry
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 which is
of finite type
universally closed (the latter means that for every the pullback is closed).
There is a classical and very practical valuative criterion of properness due Chevalley.
We say that a scheme is proper if the canonical map 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 is given by a frame , its frame of opens, and that a continuous map from to is given by an adjunction such that the inverse image function preserves finitary meets (or equivalently is a frame homomorphism, since it must preserve all joins).
Such a map is proper iff the direct image function preserves directed joins (or equivalently is Scott-continuous, or equivalently is a morphism of preframes), and also satisfies the Frobenius reciprocity-like condition that (which by itself states that the map is closed).
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.
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 ???)