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

Author Yoshihiro Maruyama



PDF
Thumbnail PDF

File

LIPIcs.CSL.2013.500.pdf
  • Filesize: 0.5 MB
  • 21 pages

Document Identifiers

Author Details

Yoshihiro Maruyama

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.CSL.2013.500

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.
Keywords
  • duality
  • monad
  • categorical topology
  • domain theory
  • convex structure

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail