
Contents
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Abstract Abstract
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13.1 Introduction 13.1 Introduction
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13.2 Preliminaries on Formal Covers 13.2 Preliminaries on Formal Covers
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13.2.1 Join formal covers 13.2.1 Join formal covers
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13.3 Locally Compact Formal Covers 13.3 Locally Compact Formal Covers
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13.4 Exponentiation 13.4 Exponentiation
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13.4.1 The exponential formal cover 13.4.1 The exponential formal cover
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13.4.2 Application and abstraction 13.4.2 Application and abstraction
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13.5 Any Exponentiable Formal Cover is Locally Compact 13.5 Any Exponentiable Formal Cover is Locally Compact
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Acknowledgements Acknowledgements
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References References
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13 PREDICATIVE EXPONENTIATION OF LOCALLY COMPACT FORMAL TOPOLOGIES OVER INDUCTIVELY GENERATED TOPOLOGIES
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Published:October 2005
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Abstract
Martin Hyland first classified the exponentiable locales as the locally compact ones. This chapter proves the counterpart of Hyland's result for inductively generated formal topologies. More precisely, by adapting Hyland's proofs to the context of formal topology, it proves that locally compact formal covers are the exponentiable objects in the category of inductively generated formal covers. Then, it deduces the analogous result for inductively generated formal topologies (that is, formal topologies with a unary positivity predicate). This demonstrates once more how impredicative reasoning can be unwound predicatively with great patience, but with the benefit of a quite explicit proof.
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