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An Elementary Theory of the Category of Locally Compact Locales
, 2003
"... 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 selfadjoint exponential # , and Scott continuity, F# = ##. ..."
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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 selfadjoint exponential # , and Scott continuity, F# = ##.
An Elementary Theory of Various Categories of Spaces in Topology
, 2005
"... In Abstract Stone Duality the topology on a space X is treated, not as an infinitary lattice, but as an exponential space ΣX. This has an associated lambda calculus, in which monadicity of the selfadjunction Σ − a Σ − makes all spaces sober and gives subspaces the subspace topology, and the Euclid ..."
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In Abstract Stone Duality the topology on a space X is treated, not as an infinitary lattice, but as an exponential space ΣX. This has an associated lambda calculus, in which monadicity of the selfadjunction Σ − a Σ − makes all spaces sober and gives subspaces the subspace topology, and the Euclidean principle Fσ ∧ σ = F> ∧ σ makes Σ the classifier for open subspaces. Computably based locally compact locales provide the leading model for these axioms, although the methods are also applicable to CCDop (constructively completely distributive lattices). In this paper we recover the textbook theories, using the additional axiom that the subcategory of overt discrete objects have a coreflection, the “underlying set ” functor. This subcategory is then a topos, and the whole category is characterised in the minimal situation as that of locally compact locales over that topos. However, by adding further axioms regarding the existence of equalisers and injectivity of Σ, we find the category of sober spaces or of locales over the topos as a reflective subcategory, whilst the whole category is cartesian closed and has all finite limits and colimits.
Foundations for abstract forcing
, 712
"... Abstract: The foundations of forcing theory are reworked to streamline the presentation and to show how the most basic results are applicable in very general contexts. ..."
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Abstract: The foundations of forcing theory are reworked to streamline the presentation and to show how the most basic results are applicable in very general contexts.
MATHEMATICAL LOGIC QUARTERLY
, 2007
"... The axiomofchoice and the law of excluded middle in weak set theories ..."
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