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DOI: 10.4230/LIPIcs.CSL.2013.500

Categorical Duality Theory: With Applications to Domains, Convexity, and the Distribution Monad

Authors: Yoshihiro Maruyama

Published in: LIPIcs, Volume 23, Computer Science Logic 2013 (CSL 2013)

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.

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Yoshihiro Maruyama. Categorical Duality Theory: With Applications to Domains, Convexity, and the Distribution Monad. In Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 23, pp. 500-520, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{maruyama:LIPIcs.CSL.2013.500,
  author =	{Maruyama, Yoshihiro},
  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 =	{Ronchi Della Rocca, Simona},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2013.500},
  URN =		{urn:nbn:de:0030-drops-42168},
  doi =		{10.4230/LIPIcs.CSL.2013.500},
  annote =	{Keywords: duality, monad, categorical topology, domain theory, convex structure}
}

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Categorical Duality Theory: With Applications to Domains, Convexity, and the Distribution Monad

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