Notes on sublocales and dissolution

Abstract

The dissolution (introduced by Isbell in [3], discussed by Johnstone 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 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 ^cT(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 ultraparacompactness) of S(L) which becomes (we hope) substantially more transparent.