Notes on sublocales and dissolution (Preprint)

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Type: Preprint
National /International: International
Title: Notes on sublocales and dissolution
Publication Date: 2023-08-08
Authors: - Jorge Picado
- Ales Pultr

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
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