A discrete group $G$ acts properly discontinuously on a manifold $M$ if the set $\{g\in G\mid gK\cap K\neq \emptyset \}$ is finite for every compact $K\subset M$.

Is there a more abstract characterisation of this property?

I am looking for something that looks like a diagram, a functor or anything vaguely related (to homological algebra, category theory). I have had a look at the relevant pages on nlab, but, although there is a description for a proper map of schemes, I can't think of any formal way of characterising a proper map of topological spaces.

EDIT: I think my question was somewhat imprecise. (Apologies.)

How can the definition of a proper action be defined for group objects in an arbitrary category such that it reduces to a proper action of a Lie group in the category of smooth manifolds and a properly discontinuous action of a group (object) in the category of sets?