2016 Fiscal Year Research-status 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 / monad algebras / powerspace |
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| Outline of Annual Research Achievements |
The purpose of this research is to investigate topological algebra for quasi-Polish (countably based completely quasi-metrizable) spaces. I have been mainly focusing on characterizations of the topological semilattices that arise as the Eilenberg-Moore algebras of powerspace monads on the category of quasi-Polish spaces.
The lower powerspace monad is related via Stone duality to the "possibility" operator in modal logic, and the algebras of the monad correspond to certain topological join semilattices. The lower powerspace monad and its algebras in the category of quasi-Polish spaces is now very well understood.The upper powerspace monad corresponds to "necessity" in modal logic, and the algebras correspond to certain topological meet semilattices. We have made much progress on the quasi-Polish algebras of the upper powerspace, and now have topological characterizations of its algebras, but the characterization is still not as clear as the case for the lower powerspace algebras.
We also showed that the upper and lower powerspace monads on quasi-Polish spaces distribute (in the sense of Beck), and found the topological property necessary for this distributivity (they do not distribute on arbitrary spaces, contrary to the case for locales). The composition of the upper and lower powerspace monads yields the double powerspace monad, and the algebras of this monad correspond to certain quasi-Polish distributive lattices, which are in fact frames. We are currently focusing on better characterizations of these quasi-Polish frames, but this appears to be a very challenging problem.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
A full characterization of the Eilenberg-Moore algebras of the double powerspace monad on quasi-Polish spaces (which are certain quasi-Polish frames) hinges on a characterization of the algebras of the upper powerspace monad (which are certain quasi-Polish meet-semilattices). A characterization of the latter has been rather challenging, but we have made important progress over the past year. Earlier investigations into algebras of the upper powerspace monad depended on the structure of the compact subsets of the space, but we have been able to apply various completeness properties of quasi-Polish spaces to reduce the problem to more finitary aspects of the meet-semilattice structure, which has greatly simplified the problem. The remaining problem seems related to Pontryagin duality, the proof of which is rather complicated, but there are simpler variations that occur in topological semilattice theory that we are currently investigating for hints on how to attack the problem.
Dually, we have identified many topological spaces whose frame of opens (endowed with the Scott-topology) is a quasi-Polish frame. These spaces are typically not countably based (hence not quasi-Polish) and are notorious for being difficult to analyze, but we are currently investigating methods to apply the developing theory of quasi-Polish frames to analyze the spaces via Stone duality. Characterizing these "dual" spaces is also a challenging problem, but it opens up many interesting applications for the theory being developed during this research project.
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| Strategy for Future Research Activity |
Currently, the two major goals in sight is to find a more direct characterization of the quasi-Polish meet-semilattices that are algebras of the upper powerspace monad, and to characterize the "dual" spaces whose frames are quasi-Polish. Towards this goal, we are investigating the proofs of related dualities, such as Pontryagin duality and its variations in topological semilattice theory such as Hoffmann-Mislove-Stralka duality. However, this problem appears to be rather challenging, and so we are also investigating other applications in case reasonable progress cannot be made over the next year.
In particular, we are interested in applying the theory being developed to give semantics of modal logic and non-deterministic systems. In order to develop these applications it is important to find constructive proofs of the characterizations we already have, and we are currently working on finding constructive proofs using geometric logic by applying methods from point-free topology.
We also plan to further investigate presentations of the frame of opens of a space and its algebraic structure in terms of generators and relations. It is known from results by R. Heckmann that quasi-Polish spaces dually correspond to frames that have a countable presentation. We have made some progress on generalizing Heckmann's result to certain non-countable presentations of quasi-Polish spaces. An interesting problem is to extend this to include presentations of algebraic structure on quasi-Polish spaces, similar to the way that Jung-Moshier-Vickers provided presentations of dcpo algebras.
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| Causes of Carryover |
Only 13万 was left over, and it was deemed to be more useful to use it in the following year along with the additional funds than try to spend it all this year.
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| Expenditure Plan for Carryover Budget |
The remaining funds will be combined with this year's funds for conference expenses, inviting researchers, and other supplies.
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