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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CSL.2013.500
URN: urn:nbn:de:0030-drops-42168
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2013/4216/
Maruyama, Yoshihiro
Categorical Duality Theory: With Applications to Domains, Convexity, and the Distribution Monad
Abstract
Utilising and expanding concepts from categorical topology and algebra, we contrive a moderately general theory of dualities between algebraic, point-free spaces and set-theoretical, point-set spaces, which encompasses infinitary Stone dualities, such as the well-known duality between frames (aka. locales) and topological spaces, and a duality between \sigma-complete Boolean algebras and measurable spaces, as well as the classic finitary Stone, Gelfand, and Pontryagin dualities. Among different applications of our theory, we focus upon domain-convexity duality in particular: from the theory we derive a duality between Scott's continuous lattices and convexity spaces, and exploit the resulting insights to identify intrinsically the dual equivalence part of a dual adjunction for algebras of the distribution monad; the dual adjunction was uncovered by Bart Jacobs, but with no characterisation of the induced equivalence, which we do give here. In the Appendix, we place categorical duality in a wider context, and elucidate philosophical underpinnings of duality.
BibTeX - Entry
@InProceedings{maruyama:LIPIcs:2013:4216,
author = {Yoshihiro Maruyama},
title = {{Categorical Duality Theory: With Applications to Domains, Convexity, and the Distribution Monad}},
booktitle = {Computer Science Logic 2013 (CSL 2013)},
pages = {500--520},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-60-6},
ISSN = {1868-8969},
year = {2013},
volume = {23},
editor = {Simona Ronchi Della Rocca},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2013/4216},
URN = {urn:nbn:de:0030-drops-42168},
doi = {10.4230/LIPIcs.CSL.2013.500},
annote = {Keywords: duality, monad, categorical topology, domain theory, convex structure}
}
Keywords: |
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duality, monad, categorical topology, domain theory, convex structure |
Collection: |
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Computer Science Logic 2013 (CSL 2013) |
Issue Date: |
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2013 |
Date of publication: |
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02.09.2013 |