eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2013-09-02
500
520
10.4230/LIPIcs.CSL.2013.500
article
Categorical Duality Theory: With Applications to Domains, Convexity, and the Distribution Monad
Maruyama, Yoshihiro
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol023-csl2013/LIPIcs.CSL.2013.500/LIPIcs.CSL.2013.500.pdf
duality
monad
categorical topology
domain theory
convex structure