<Reference List> | |
Type: | Preprint |
National /International: | International |
Title: | Notes on sublocales and dissolution |
Publication Date: | 2023-08-08 |
Authors: |
- Jorge Picado
- Ales Pultr |
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 \( \mathsf{T}(L)=\mathsf{S}(L)^{op} \) mediated by the natural embedding \( \mathfrak{c}_L=(a \mapsto \uparrow a) \) and its adjoint localic map \( \gamma_L\colon \mathsf{T}(L)\to L \). The associated image-preimage adjunction \( \gamma_L^{-1}[-]\dashv \gamma_L[-] \) between the frames \( \mathsf{T}(L) \) and \( \mathsf{TT}(L) \) is shown to coincide with the adjunction \( \mathfrak{c}_{\mathsf{T}(L)}\dashv \gamma_{\mathsf{T}(L)} \) of the second step of the assembly (tower) of \( L \). This helps to explain the role of \( \mathsf{T}(L)=\mathsf{S}(L)^{op} \) as an "almost discrete lift" (sometimes used as a sort of model of the classical discrete lift \( \mathsf{D}(L)\to L \)) 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 \( \mathsf{S}(L) \) which becomes (we hope) substantially more transparent. |
Institution: | DMUC 23-26 |
Online version: | http://www.mat.uc.pt...prints/eng_2023.html |
Download: | Not available |