Abstract
Utilising and expanding concepts from categorical topology and algebra, we contrive a moderately general theory of dualities between algebraic, pointfree spaces and settheoretical, pointset spaces, which encompasses infinitary Stone dualities, such as the wellknown duality between frames (aka. locales) and topological spaces, and a duality between \sigmacomplete 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 domainconvexity 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 = {500520},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897606},
ISSN = {18688969},
year = {2013},
volume = {23},
editor = {Simona Ronchi Della Rocca},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2013/4216},
URN = {urn:nbn:de:0030drops42168},
doi = {10.4230/LIPIcs.CSL.2013.500},
annote = {Keywords: duality, monad, categorical topology, domain theory, convex structure}
}
Keywords: 

duality, monad, categorical topology, domain theory, convex structure 
Seminar: 

Computer Science Logic 2013 (CSL 2013) 
Issue Date: 

2013 
Date of publication: 

27.08.2013 