Notes on sublocales and dissolution

Abstract

The dissolution (introduced by Isbell in [3], discussed by John-stone in [5] and later exploited by Plewe in [12, 13]) is here viewed as the relation of the geometry of L with that of the more dispersed T ( L ) = S ( L ) op mediated by the natural embedding c L = ( a 7→ ↑ a ) and its adjoint localic map γ L : T ( L ) → L . The associated image-preimage adjunction ( γ L ) − 1 [ − ] ⊣ γ L [ − ] between the frames T ( L ) and TT ( L ) is shown to coincide with the adjunction c T ( L ) ⊣ γ T ( L ) of the second step of the assembly (tower) of L . This helps to explain the role of T ( L ) = S ( L ) op as an “almost discrete lift” (sometimes used as a sort of model of the classical discrete lift DL → L ) as a dispersion going halfway to Booleanness. Consequent use of the concrete sublocales technique simplifies the reasoning. We illustrate it on the celebrated Plewe’s Theorem on ultranormality (and ultrapara-compactness) of S ( L ) which becomes (we hope) substantially more transparent.