Convergence without points

Abstract

We introduce a pointfree theory of convergence on lattices and coframes. A convergence lattice is a lattice $L$ with a monotonic map $\lim_L$ from the lattice of filters on $L$ to $L$, meant to be an abstract version of the map sending every filter of subsets to its set of limits. This construction exhibits the category of convergence spaces as a coreflective subcategory of the opposite of the category of convergence lattices. We extend this construction to coreflections between limit spaces and the opposite of so-called limit lattices and limit coframes, between pretopological convergence spaces and the opposite of so-called pretopological convergence coframes, between adherence spaces and the opposite of so-called adherence coframes, between topological spaces and the opposite of so-called topological coframes. All of our pointfree categories are complete and cocomplete, and topological over the category of coframes. Our final pointfree category, that of topological coframes, shares with the category of frames the property of being in a dual adjunction with the category of topological spaces. We show that the latter arises as a retract of the former, and that this retraction restricts to a reflection between frames and so-called strong topological coframes.