Generating sublocales by subsets and relations: a tangle of adjunctions

Abstract

Generalizing the obvious representation of a subspace Y⊆X as a sublocale in Ω(X) by the congruence {(U,V)|U∩Y=V∩Y}, one obtains the congruence {(a,b)|o(a)∩S=o(b)∩S}, first with sublocales S of a frame L, which (as it is well known) produces back the sublocale S itself, and then with general subsets S⊆L. The relation of such S with the sublocale produced is studied (the result is not always the sublocale generated by S). Further, we discuss in general the associated adjunctions, in particular that between relations on L and subsets of L and view the aforementioned phenomena in this perspective.