Joins of closed sublocales are not always a coframe

2025/10/01 by Arrieta, Igor
#06D22 #18F70 #Category Theory (math.CT) #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.2510.00987

Abstract

Given a locale L, the collection S_c(L) of joins of closed sublocales forms a frame--somewhat unexpectedly, as it is naturally embedded in the coframe of all sublocales of L, where by coframe we mean the order-theoretic dual of a frame. This construction has attracted attention in point-free topology: as a maximal essential extension in the category of frames, for its (non-)functorial properties, its relation to canonical extensions and exact filters of frames, etc. A central open question of the theory, posed by Picado, Pultr, and Tozzi in 2019, asked whether S_c(L) is always a coframe, or whether there exists a locale for which this fails. In this paper, we resolve this question in the negative by constructing a locale L such that S_c(L) is not a coframe. The main challenge in such questions lies in the difficulty of understanding exact infima in S_c(L); we circumvent this by analysing a certain separation property satisfied by S_c(L).

Citations

• Raney extensions: a pointfree theory of T₀ spaces based on canonical extension
• Canonical extensions via fitted sublocales
• Localic separation and the duality between closedness and fittedness

Related

• The coframe of D-sublocales of a locale and the T_D duality
• Canonical extensions via fitted sublocales
• Semilattice base hierarchy for frames and its topological ramifications
• A Primer on Chainmails: Structures for Point-free Connectivity
• Strictly zero-dimensional biframes and Raney extensions