A convexity space is a set X with a chosen family of subsets (called convex subsets) that is closed under arbitrary intersections and directed unions. There is a lot of interest in spaces that have both a convexity space and a topological space structure. In this paper, we study the category of topological convexity spaces and extend the Stone duality between coframes and topological spaces to an adjunction between topological convexity spaces and sup-lattices. We factor this adjunction through the category of preconvexity spaces (somtimes called closure spaces).
We show explicitly that Boolean inverse semigroups are in duality with what we term Boolean groupoids. This generalizes the classical Stone duality, which we refer to as commutative Stone duality, between generalized Boolean algebras and locally compact Hausdorff $0$-dimensional spaces.
We establish a duality between global sheaves on spectral spaces and right distributive bands. This is a sheaf-theoretical extension of classical Stone duality between spectral spaces and bounded distributive lattices. The topology of a spectral space admits a refinement, the so-called patch topology, giving rise to a patch monad on sheaves over a fixed spectral space. Under the duality just mentioned the algebras of this patch monad are shown to correspond to distributive skew lattices.
Extending the Stone Duality Theorem, we prove two duality theorems for the category ZHaus of zero-dimensional Hausdorff spaces and continuous maps. Both of them imply easily the Tarski Duality Theorem, as well as two new duality theorems for the category EDTych of extremally disconnected Tychonoff spaces and continuous maps. Also, we describe two categories which are dually equivalent to the category ZComp of zero-dimensional Hausdorff compactifications of zero-dimensional Hausdorff spaces and obtain as a corollary the Dwinger Theorem about zero-dimensional compactifications of a zero-dimensio…
We show how Stone duality can be extended from maps to relations. This is achieved by working order enriched and defining a relation from A to B as both an order-preserving function from the opposite of A times B to the 2-element chain and as a subobject of A times B. We show that dual adjunctions and equivalences between regular categories, taken in a suitably order enriched sense, extend to (framed bi)categories of relations.
Gelfand-Naimark-Stone duality establishes a dual equivalence between the category ${\sf KHaus}$ of compact Hausdorff spaces and the category ${\boldsymbol{\mathit{uba}\ell}}$ of uniformly complete bounded archimedean $\ell$-algebras. We extend this duality to the category ${\sf CReg}$ of completely regular spaces. This we do by first introducing basic extensions of bounded archimedean $\ell$-algebras and generalizing Gelfand-Naimark-Stone duality to a dual equivalence between the category ${\boldsymbol{\mathit{ubasic}}}$ of uniformly complete basic extensions and the category ${\sf C}$ of comp…
We prove three new versions of Stone Duality. The main version is the following: the category of Kolmogorov locally small spaces and bounded continuous mappings is equivalent to the category of spectral spaces with decent lumps and with bornologies in the lattices of compact (not necessarily Hausdorff) open sets as objects and spectral mappings respecting the decent lumps and satisfying a boundedness condition as morphisms as well as it is dually equivalent to the category of bounded distributive lattices with bornologies and with decent lumps of prime filters as objects and homomorphisms of b…
A common feature of many duality results is that the involved equivalence functors are liftings of hom-functors into the two-element space resp. lattice. Due to this fact, we can only expect dualities for categories cogenerated by the two-element set with an appropriate structure. A prime example of such a situation is Stone's duality theorem for Boolean algebras and Boolean spaces,the latter being precisely those compact Hausdorff spaces which are cogenerated by the two-element discrete space. In this paper we aim for a systematic way of extending this duality theorem to categories including …
The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this paper, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski's observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean …
Classic work of Pierce and Dauns-Hofmann shows that biregular rings are dual to simple ring bundles over Stone spaces. We extend this duality to Steinberg rings, a purely algebraic generalisation of Steinberg algebras, and ringoid bundles over ample groupoids. We base this largely on an even more general extension of Lawson's noncommutative Stone duality, specifically between Steinberg semigroups, a generalisation of Boolean inverse semigroups, and category bundles over ample groupoids.
A De Vries-type Duality Theorem for Locally Compact Spaces -- II
本文给出了第一部分的方法和结果以及M。Stone，H. de Vries，P. Roeper给出。特别是：得到了Stone对偶定理的一些推广，证明了局部切触布尔代数的一个完备定理，给出了Birkhoff问题72的Ponomarev解的一个直接证明，并描述了与（零维）Eberlein紧空间是余绝对的空间.
We prove versions of the spectral adjunction, a Stone-type duality and Hofmann-Lawson duality for locally small spaces with bounded continuous mappings.
The term Stone-type duality often refers to a dual equivalence between a category of lattices or other partially ordered structures on one side and a category of topological structures on the other. This paper is part of a larger endeavour that aims to extend a web of Stone-type dualities from ordered to metric structures and, more generally, to quantale-enriched categories. In particular, we improve our previous work and show how certain duality results for categories of [0,1]-enriched Priestley spaces and [0,1]-enriched relations can be restricted to functions. In a broader context, we inves…
Some Generalizations of Fedorchuk Duality Theorem -- I
推广V. V. Fedorchuk的对偶定理，证明了以下四类情形的Stone型对偶定理：它们都以局部紧的T2空间为对象，它们的态射分别是连续骨架映射、拟开完美映射、开映射、开完美映射.特别地，证明了所有紧T2空间及其之间的所有开映射范畴的Stone型对偶定理.我们还得到了这四个范畴的等价定理。这些定理的版本的满子范畴的这些范畴作为对象的所有局部紧连通T2空间制定以及。
It has been known since the work of Duskin and Pelletier four decades ago that KH^op, the category opposite to compact Hausdorff spaces and continuous maps, is monadic over the category of sets. It follows that KH^op is equivalent to a possibly infinitary variety of algebras V in the sense of Slominski and Linton. Isbell showed in 1982 that the Lawvere-Linton algebraic theory of V can be generated using a finite number of finitary operations, together with a single operation of countably infinite arity. In 1983, Banaschewski and Rosicky independently proved a conjecture of Bankston, establishi…
We unify several extensions of the classic Stone duality due to Grätzer, Hoffman-Lawson and Jung-Sünderhauf. Specifically we show that U-bases of locally compact sober spaces are dual to <-distributive v-predomains, where < is a transitive relation representing compact containment.
Under a general categorical procedure for the extension of dual equivalences as presented in this paper's predecessor, a new algebraically defined category is established that is dually equivalent to the category $\bf LKHaus$ of locally compact Hausdorff spaces and continuous maps, with the dual equivalence extending a Stone-type duality for the category of extremally disconnected locally compact Hausdorff spaces and continuous maps. The new category is then shown to be isomorphic to the category $\bf CLCA$ of complete local contact algebras and suitable morphisms. Thereby, a new proof is pres…
We investigate an enriched-categorical approach to a field of discrete mathematics. The main result is a duality theorem between a class of enriched categories (called $\overline{\mathbb{Z}}$- or $\overline{\mathbb{R}}$-categories) and that of what we call ($\overline{\mathbb{Z}}$- or $\overline{\mathbb{R}}$-) extended L-convex sets. We introduce extended L-convex sets as variants of certain discrete structures called L-convex sets and L-convex polyhedra, studied in the field of discrete convex analysis. We also introduce homomorphisms between extended L-convex sets. The theorem claims that th…
