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Author Maruyama, Yoshihiro
Source CiteSeerX
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Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Distribution Monad ♦ Categorical Duality Theory ♦ Dual Adjunction ♦ General Theory ♦ Continuous Lattice ♦ Complete Boolean Algebra ♦ Well-known Duality ♦ Infinitary Stone Duality ♦ Induced Equivalence ♦ Pontryagin Duality ♦ Measurable Space ♦ Different Application ♦ Point-set Space ♦ Classic Finitary Stone ♦ Topological Space ♦ Domain-convexity Duality ♦ Convexity Space ♦ Wider Context ♦ Categorical Duality ♦ Dual Equivalence Part ♦ Philosophical Underpinnings ♦ Bart Jacob ♦ Categorical Topology ♦ Point-free Space
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 σ-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.
Educational Role Student ♦ Teacher
Age Range above 22 year
Educational Use Research
Education Level UG and PG ♦ Career/Technical Study