The Becker--Kechris theorem states that every Borel action of a Polish group may be made continuous, and moreover gives a characterization (in some sense) of all possible such topological realizations. We give a new proof of the Becker--Kechris theorem, with a more abstract and topological flavor than the original proof, based entirely on algebraic properties of Baire category quantifiers. Using this proof method, we obtain several new extensions of the result: to n-ary relations; to groupoids; to non-Hausdorff spaces; and even to point-free "spaces".