We discuss the problem of characterizing the property of a Grothendieck topos to satisfy a given `geometric' invariant as a property of its sites of de nition, and indicate a set of general techniques for establishing such criteria. We then apply our methodologies to speci c invariants, notably including the property of a Grothendieck topos to be localic (resp. atomic, locally connected, equiva...

Preframes (directed complete posets with finite meets that distribute over the directed joins) are the algebras for an infinitary essentially algebraic theory, and can be presented by generators and relations. This result is combined with a general argument concerning categories of commutative monoids to give a very short proof of the localic Tychonoff Theorem. It is also shown how frames can b...

The localic definitions of cosheaves, connectedness and local connectedness are transferred from impredicative topos theory to predicative formal topology. A formal topology is locally connected (has base of connected opens) iff it has a cosheaf π0 together with certain additional structure and properties that constrain π0 to be the connected components cosheaf. In the inductively generated cas...