The foundations of locale theory

Abstract

Locale theory (or point-free topology) may be regarded as the study of topology on a lattice-theoretic foundation. Instead of topological spaces, the objects of study are called frames (or locales); these are complete lattices which satisfy a particular infinite distributive law. Frames and their homomorphisms offer one perspective of point-free topology, another perspective is given by locales and their morphisms (localic maps). The theory of locales generalises sober space topology; moreover, there is a categorical duality between the so-called spatial frames and sober spaces. Locale theory genuinely offers new insights and results in topology, for instance, sublocales (i.e. generalised subspaces) are better behaved than their classical counterparts: every locale contains a least dense sublocale – a result which has no analogue in point-set topology – this has the consequence that there are more point-free spaces than classical spaces, i.e. there exist locales (the non-spatial ones) which do not arise from topological spaces. Since frames are algebras, frames may be presented by generators and relations; in our case the generators form a meet-semilattice and the relations are encoded in a coverage on the generators – a meet-semilattice equipped with a coverage is called a site, and every site canonically freely generates a frame (for example, the frame of ideals of a distributive lattice is just a special case of a freely generated frame over a site). Compactness of locales is a point-free invariant, and there is a localic Kuratowski-Mrówka Theorem which characterises compact locales via closed projections from a binary coproduct of frames – this offers a categorical (or extrinsic) characterisation of localic compactness.