2017 Fiscal Year Annual Research Report
Research on complete quasi-metric spaces with algebraic structure
| Project/Area Number |
15K15940
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| Research Institution | Kyoto University |
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Principal Investigator |
ディブレクト マシュー 京都大学, 人間・環境学研究科, 特定講師 (20623599)
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| Project Period (FY) |
2015-04-01 – 2018-03-31
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| Keywords | quasi-Polish space / topological algebra / semilattices / powerspace |
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| Outline of Annual Research Achievements |
Basic results for the upper, lower, and double powerspace monads on the category of quasi-Polish spaces that were obtained through joint work with T. Kawai have been written up and submitted to an international journal (a preprint is available on the arXiv: https://arxiv.org/pdf/1709.06226.pdf ). It is known from previous work by A. Schalk that algebras of the upper powerspace monad correspond to certain topological meet semilattices, and that algebras of the lower powerspace monad correspond to certain topological join semilattices. We showed that a topological property known as “consonance” (a property held by all quasi-Polish spaces) is precisely what is needed for the upper and lower powerspaces to commute with each other.
The double powerspace can be defined as the composition of the upper and lower powerspaces, and the commutativity of the upper and lower powerspaces implies that the quasi-Polish algebras of the double powerspace monad are certain topological distributive lattices. In fact, they are a kind of topological “frame”. Some progress has been made in characterizing these “quasi-Polish frames”, although much work remains. These characterizations were not included in the preprint, although the preprint does explicitly show how this works in the “dual” case, namely that the frame of opens of a quasi-Polish space equipped with the Scott-topology is an algebra of both the upper and lower powerspace monads.
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