Chu duality theory and coalgebraic representation of quantum symmetries |
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Affiliation: | 1. Department of Mathematics, Endenicher Allee 60, 53115 Bonn, Germany;2. Department of Mathematics, 1650 Bedford Avenue, 11225 Brooklyn, United States;3. Mathematics Program, The Graduate Center, CUNY, 365 Fifth Avenue, 10016 New York, United States;1. Université catholique de Louvain, Institut de Recherche en Mathématique et Physique, 1348 Louvain-la-Neuve, Belgium;2. Università degli Studi di Padova, Dipartimento di Matematica “Tullio Levi-Civita”, 35121 Padova, Italy;1. Department of Mathematics, University of York, York YO10 5DD, UK;2. Department of Mathematics, University of Waikao, Hamilton 3216, New Zealand |
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Abstract: | Building upon Vaughan Pratt's work on applications of Chu space theory to Stone duality, we develop a general theory of categorical dualities on the basis of Chu space theory and closure conditions, which encompasses a variety of dualities for topological spaces, convex spaces, closure spaces, and measurable spaces (some of which are new duality results on their own). It works as a general method to generate analogues of categorical dualities between frames (locales) and topological spaces beyond topology, e.g., for measurable spaces, convex spaces, and closure spaces. After establishing the Chu duality theory, we apply the state-observable duality between quantum lattices and closure spaces to coalgebraic representations of quantum symmetries, showing that the quantum symmetry groupoid fully embeds into a purely coalgebraic category, i.e., the category of Born coalgebras, which refines, through the quantum duality that follows from Chu duality theory, Samson Abramsky's fibred coalgebraic representations of quantum symmetries (which, in turn, builds upon his Chu representations of symmetries). |
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Keywords: | 18F70 18E20 16T15 81Q60 |
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