Joins in the frame of nuclei are hard to describe explicitly because a pointwise join of a set of closure operators on a complete lattice fails to be idempotent in general. We calculate joins of nuclei as least fixed points of inflationary operators on prenuclei. Using a recent fixed-point theorem due to Pataraia, we deduce an induction principle for joins of nuclei. As an illustration of the t...

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...