; TeX output 2003.08.11:2113 C Ghtml: html: Gz html: html:4 UDt G G cmr17Subspaces7tinAbstractStoneDualitqy /cXQ cmr12PraulTVaylorfj August11,2003 U*t: cmbx9Abstract э&)o cmr9Byec+j cmti9abstract?*Stonedualit9ywemeanthatthetopAologyorcontrav|rariantpAowersetfunctor, seen5asaself-adjoin9texpAonential-=Z cmr5(0n cmsy5 j)Nonsomecategory:,>ismonadic.~FUsingBeck'stheorem, thismeansthatcertainequalisersexistandcarrythesubspacetopAology:.JThesesubspacesare encoAdedTb9yidempoten9tsthatplayarolesimilartothatofnucleiinloAcaletheory:.&P9arXezshowedthatanyelementarytopAoshasthisduality:,andweproveitintuitionistically forTthecategoryofloAcallycompactlocales.&TheypapAerislargelyconcernedwiththeconstructionofsuc9hacategoryoutofonethat merely8NhaspAo9wers8Nofsomexedob ject._ItbuildsonSoberYSpacesandContinuations, wheretherelatedbutw9eakernotionofabstractsobriet9ywasconsidered.Theconstructionis donerstb9yformallyadjoiningcertainequalisersthat-=( j)ULtakestocoAequalisers,6thenusing Eilen9bAerg{Moore0algebras,andnallypresentedasalambAdacalculussimilartotheaxiomof comprehensionTinsettheory:.&Thecomprehensioncalculushasanormalisationtheorem,N1b9ywhicheverytypAecanbe em9bAeddedBasasubspaceofatypAeformedwithoutcomprehension,M,andtermsalsonormalise inasimplew9ay:.ThesymbAolicandcategoricalstructuresaretherebyshowntobAeequiv|ralent.&Finally:,sumsLandcertainquotien9tsareconstructedusingthecomprehensioncalculus, givingTanextensiv9ecategory:.9Ys1+OK`y
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