# SUBSPACES IN ABSTRACT STONE DUALITY

@inproceedings{Taylor2002SUBSPACESIA, title={SUBSPACES IN ABSTRACT STONE DUALITY}, author={Paul Taylor}, year={2002} }

B yabstract Stone duality we mean that the topology or contravariant powerset functor, seen as a self-adjoint exponential Σ (−) on some category, is monadic. Using Beck's theorem, this means that certain equalisers exist and carry the subspace topology. These subspaces are encoded by idempotents that play a role similar to that of nuclei in locale theory. Pare showed that any elementary topos has this duality, and we prove it intuitionistically for the category of locally compact locales. The…

## 33 Citations

An Elementary Theory of Various Categories of Spaces in Topology

- Mathematics
- 2005

In Abstract Stone Duality the topology on a space X is treated, not as an innitary lattice, but as an exponential space X . This has an associated lambda calculus, in which monadicity of the…

Non-Artin Gluing in Recursion Theory and Lifting in Abstract Stone Duality

- Mathematics
- 2003

Stone duality is a radical reformulation of general topology, in which the topology on a space X is not considered as a set carrying an infinitary lattice structure, but as another space that’s the…

Computably Based Locally Compact Spaces

- MathematicsLog. Methods Comput. Sci.
- 2006

This paper uses the full subcategory of overt discrete objects of ASD to translate computable bases for classical spaces into objects in the ASD calculus, and shows this subcategory to be equivalent to a notion of computable basis for locally compact sober spaces or locales.

The Dedekind reals in abstract Stone duality

- MathematicsMathematical Structures in Computer Science
- 2009

The core of the paper constructs the real line using two-sided Dedekind cuts, and shows that the closed interval is compact and overt, where these concepts are defined using quantifiers.

Geometric and Higher Order Logic in terms of Abstract Stone Duality

- Mathematics
- 2000

The contravariant powerset and its generalisations X to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers satisfy the Euclidean…

A Coalgebraic Approach to Dualities for Neighborhood Frames

- MathematicsArXiv
- 2021

An endofunctor is constructed on the category of complete and atomic Boolean algebras that is dual to the double powerset functor on Set to show that Thomason duality for neighborhood frames can be viewed as an algebra-coalgebra duality.

SOBER SPACES AND CONTINUATIONS

- Mathematics
- 2002

A topological space is sober if it has exactly the points that are dictated by its open sets. We explain the analogy with the way in which computational values are determined by the observations that…

Bases for algebras over a monad

- MathematicsArXiv
- 2020

It is shown that the notion of a basis can be extended to algebras of arbitrary monads on arbitrary categories and is able to recover known constructions from automata theory, namely the so-called canonical residual finite state automaton.

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Stone duality is a radical reformulation of general topology, in which the topology on a space X is not considered as a set carrying an infinitary lattice structure, but as another space that’s the…

SOBER SPACES AND CONTINUATIONS

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A topological space is sober if it has exactly the points that are dictated by its open sets. We explain the analogy with the way in which computational values are determined by the observations that…

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