Regular closure operators

In an 〈E,M 〉-category X for sinks, we identify necessary conditions for Galois connections from the power collection of the class of (composable pairs) of morphisms in M to factor through the “lattice” of all closure operators on M , and to factor through certain sublattices. This leads to the notion of regular closure operator. As one byproduct of these results we not only arrive (in a novel way) at the Pumplün–Röhrl polarity between collections of morphisms and collections of objects in such a category, but obtain many factorizations of that polarity as well. (One of these factorizations constituted the main result of an earlier paper by the same authors). Another byproduct is the clarification of the Salbany construction (by means of relative dominions) of the largest idempotent closure operator that has a specified class of X -objects as separated objects. The same relation that is used in Salbany’s relative dominion construction induces classical regular closure operators as described above. Many other types of closure operators can be obtained by this technique; particular instances of this are the idempotent and modal closure operators that in a Grothendieck topos correspond to the Grothendieck topologies.