Priestley-type dualities for partially ordered structures. (English) Zbl 1343.18005

Caramello, Olivia

Ann. Pure Appl. Logic 167, No. 9, 820-849 (2016).

The well-known Priestley duality says that the category of bounded distributive lattices and bounded homomorphisms is equivalent to the dual categories of spectral spaces, of Priestley spaces, and of pairwise Stone spaces.
The article illustrates a general topos-theoretic framework to generate dualities between categories of ordered structures and categories of ordered Stone spaces. To recover the classical duality, and to generate new similar ones, the article analyses the patch topology construction on one side, and the free Boolean algebras on Priestley spaces, which correspond to the Boolean algebra of clopen sets of the Priestley space associated to a distributive lattice.
Although the framework requires a non elementary knowledge of the theory of Grothendieck toposes, Section 2 in the paper contains an in-depth discussion about the construction of free structures via syntactic categories. The illustration contains useful results for a specialist in the field, being at the same time accessible to a non expert.
Section 3 recovers the classical Priestley duality from the framework, and analyses its topos-theoretic meaning, which clarifies how the algebraic and topological aspects interact.
By observing how the classical duality arises from the abstract topos-theoretic perspective, the paper introduces a natural generalisation, which eventually leads to a method to synthesise dualities of the same kind. Here, of the same kind, or Priestley-type, as in the title, should be intended in an abstract sense, which becomes clear after reading Section 4.
In fact, as Section 5 shows, many Priestley-type dualities can be obtained by appropriate instances of the framework. As a matter of fact, the presented examples, despite their significance by their own, are nothing but a small sample of what Caramello’s framework allows to generate, which confirms the strength of the approach.
In my humble opinion, I would suggest consulting [O. Caramello, “The unification of mathematics via topos theory”, Preprint, arXiv:1006.3930] to the reader interested in developing results along the lines of the paper. In fact, the general technique behind the scenes of the article is Caramello’s toposes-as-bridges, which, although not explicitly mentioned, and inessential to understand the content of the paper, is the origin of the framework and its foundation.

Reviewer: Marco Benini (Buccinasco)

MSC:

18B25	Topoi
18B35	Preorders, orders, domains and lattices (viewed as categories)
03G10	Logical aspects of lattices and related structures
03G30	Categorical logic, topoi

Keywords:

Priestley duality; distributive lattice; Stone space; free structure; ordered structure; Grothendieck topos

References:

[1]	Beke, T.; Karazeris, P.; Rosický, J., When is flatness coherent?, Comm. Algebra, 33, 6, 1903-1912 (2005) · Zbl 1086.18008
[2]	Caramello, O., Lattices of theories, (Theories, Sites, Toposes: Relating and Studying Mathematical Theories Through Topos-Theoretic ‘Bridges’ (2016), Oxford University Press), 2009
[3]	Caramello, O., The unification of Mathematics via Topos Theory (2010)
[4]	Caramello, O., A topos-theoretic approach to Stone-type dualities (2011)
[5]	Caramello, O., A general method for building reflections, Appl. Categ. Struct., 22, 1, 99-118 (2014) · Zbl 1338.18014
[6]	Coquand, T.; Lombardi, H., Hidden constructions on abstract algebra: Krull dimension of distributive lattices and commutative rings, (Fontana, M.; etal., Commutative Ring Theory and Applications. Commutative Ring Theory and Applications, Lect. Notes Pure Appl. Math., vol. 231 (2002)), 477-499 · Zbl 1096.13507
[7]	Cornish, W. H., On H. Priestley’s dual of the category of bounded distributive lattices, Mat. Vesnik, 12(27), 4, 329-332 (1975) · Zbl 0344.06007
[8]	Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M. W.; Scott, D. S., Continuous Lattices and Domains, Encyclopedia Math. Appl., vol. 93 (2003), Cambridge University Press
[9]	van Gool, S., On sheaves and duality (2014), Radboud Universiteit Nijmegen, available at
[10]	Hochster, M., Prime ideal structure in commutative rings, Trans. Amer. Math. Soc., 142, 43-60 (1969) · Zbl 0184.29401
[11]	Hodges, W., Model Theory, Encyclopedia Math. Appl., vol. 42 (1993), Cambridge University Press
[12]	Johnstone, P. T., Sketches of an Elephant: A Topos Theory Compendium, vols. 1-2, Oxford Logic Guides, vols. 43-44 (2002), Oxford University Press
[13]	Johnstone, P. T., Stone Spaces, Cambridge Stud. Adv. Math., vol. 3 (1982), Cambridge University Press · Zbl 0499.54001
[14]	Peremans, W., Embedding a distributive lattice into a Boolean algebra, Indag. Math., 19, 73-81 (1957) · Zbl 0077.03801
[15]	Priestley, H. A., Representation of distributive lattices by means of ordered Stone spaces, Bull. Lond. Math. Soc., 2, 186-190 (1970) · Zbl 0201.01802
[16]	Priestley, H. A., Ordered topological spaces and the representation of distributive lattices, Proc. Lond. Math. Soc., 24, 3, 507-530 (1972) · Zbl 0323.06011
[17]	Stone, M. H., The theory of representations for Boolean algebras, Trans. Amer. Math. Soc., 40, 1, 37-111 (1936) · JFM 62.0033.04
[18]	Stone, M. H., Topological representations of distributive lattices and Brouwerian logics, Čas. Pěst. Math. Fys., 67, 1, 1-25 (1938) · JFM 63.0830.01
[19]	Tholen, W., Ordered topological structures, Topology Appl., 156, 12, 2148-2157 (2009) · Zbl 1171.54025