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, one discusses in general the associated adjunctions, in particular that of relations on L and subsets of L and views the aforementioned phenomena in this perspective.