• Corpus ID: 10318276

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… 
An Elementary Theory of Various Categories of Spaces in Topology
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
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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
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  • P. Taylor
  • Mathematics
    Log. Methods Comput. Sci.
  • 2006
TLDR
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
TLDR
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.
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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
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TLDR
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
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
TLDR
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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References

SHOWING 1-10 OF 72 REFERENCES
Non-Artin Gluing in Recursion Theory and Lifting in Abstract Stone Duality
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
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
An Elementary Theory of the Category of Locally Compact Locales
The category of locally compact locales over any elementary topos is characterised by means of the axioms of abstract Stone duality (monadicity of the topology, considered as a self-adjoint
Categories of Partial Maps
Strong functors and monoidal monads
In [4] we proved that a commutative monad on a symmetric monoidal closed category carries the structure of a symmetric monoidal monad ([4], Theorem 3.2). We here prove the converse, so that, taken
Categorical Structure of Continuation Passing Style
This thesis attempts to make precise the structure inherent in Continuation Passing Style (CPS). We emphasize that CPS translates λ-calculus into a very basic calculus that does not have functions as
Monads on symmetric monoidal closed categories
This note is concerned with "categories with internal horn and | and we shall use the terminology from the paper [2] by EIL~.NBERG and Kv.Imy. The result proved may be stated briefly as follows : a
Locales are Not Pointless
TLDR
An axiomatic theory is developed that can be interpreted soundly in two ways, using either lower or upper powerlocales, so that pairs of separate results can be proved as single formal theorems.
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