Deriving dualities in pointfree topology from Priestley duality. (English) Zbl 1523.18010
In pointfree topology there is a well-known dual adjunction between the category \(\mathsf{Top}\)of topological spaces and continuous maps and the category \(\mathsf{Frm}\)of frames and frame homomorphisms, restricting to a dual equivalence between the category \(\mathsf{Sob}\)of sober spaces and the category \(\mathsf{SFrm}\)of spacial frames. Further restrictions yield the following classical results.
This paper aims to provide a different perspective on these dualities by utilizing Priestley duality [H. A. Priestley, Bull. Lond. Math. Soc. 2, 186–190 (1970; Zbl 0201.01802); Proc. Lond. Math. Soc. (3) 24, 507–530 (1972; Zbl 0323.06011)], which establishes a dual equivalence between the category \(\mathsf{DLat}\)of bounded distributive lattices and bounded lattice homomorphisms and \(\mathsf{Pries}\)of Priedtley spaces and Priestley morphisms.
The synopsis of the paper goes as follows.
- ●
- Hofmann-Lawson duality between the category \(\mathsf{ConFrm}\)of continuous frames and proper frame homomorphisms and the category \(\mathsf{LKSob}\)of locally compact spaces and proper maps [K. H. Hofmann and J. D. Lawson, Trans. Am. Math. Soc. 246, 285–310 (1978; Zbl 0402.54043)].
- ●
- A dual equivalence between the full subcategory \(\mathsf{StCFrm}\)of \(\mathsf{ConFrm}\)consisting of stably continuous frames and the full subcategory \(\mathsf{StLKSp}\)of \(\mathsf{LKSob}\)consisting of stably locally compact spaces, which further restricts to a dual equivalence between the full subcategory \(\mathsf{StKFrm}\)of stably compact frames and \(\mathsf{StKSp}\)of stably compact spaces [B. Banaschewski, Lect. Notes Math. 871, 1–11 (1981; Zbl 0461.06010); G. Gierz and K. Keimel, Houston J. Math. 3, 207–224 (1977; Zbl 0359.06015); P. T. Johnstone, J. Pure Appl. Algebra 22, 229–247 (1981; Zbl 0445.18005); H. Simmons, Topology Appl. 13, 201–223 (1982; Zbl 0484.18005)].
- ●
- Isbell duality between the full subcategory \(\mathsf{KRFrm}\)of \(\mathsf{Frm}\)consisting of compact regular frames and the full subcategory \(\mathsf{KHaus}\)of \(\mathsf{Top}\)consisting of compact Hausdorff spaces [J. R. Isbell, Math. Scand. 31, 5–32 (1972; Zbl 0246.54028)].
This paper aims to provide a different perspective on these dualities by utilizing Priestley duality [H. A. Priestley, Bull. Lond. Math. Soc. 2, 186–190 (1970; Zbl 0201.01802); Proc. Lond. Math. Soc. (3) 24, 507–530 (1972; Zbl 0323.06011)], which establishes a dual equivalence between the category \(\mathsf{DLat}\)of bounded distributive lattices and bounded lattice homomorphisms and \(\mathsf{Pries}\)of Priedtley spaces and Priestley morphisms.
The synopsis of the paper goes as follows.
- § 2
- introduces the categories of frames and spaces of interest, presenting the relevant dualities.
- § 3
- addresses Priestley duality and its restriction to frames.
- § 4
- characterizes spacial frames in the language of Priestley duality, connecting the associated Priestley spaces with sober spaces.
- § 5
- further restricts this correspondence to continuous frames, their associated Priestley spsces, and locally compact sober spaces.
- § 6
- derives the duality between stably continuous frames and stably locally compact spaces by describing stability in the language of Priestley spaces. This gives a new proof of the duality between stably compact frames and stably compact spaces.
- § 7
- describes regularity in the language of Priestley spaces, providing an alternative proof of Isbell duality.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18F70 | Frames and locales, pointfree topology, Stone duality |
| 06D22 | Frames, locales |
| 06D50 | Lattices and duality |
| 06E15 | Stone spaces (Boolean spaces) and related structures |
| 54D30 | Compactness |
| 54D45 | Local compactness, \(\sigma\)-compactness |
Keywords:
pointfree topology; spatial frame; continuous frame; stably compact frame; compact regular frame; sober space; locally compact space; stably compact space; compact Hausdorff space; Priestley dualityCitations:
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