Hausdorff Coalgebras
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Hausdorff Coalgebras Dirk Hofmann1,2
· Pedro Nora1,2
Received: 14 August 2019 / Accepted: 23 March 2020 © Springer Nature B.V. 2020
Abstract As composites of constant, finite (co)product, identity, and powerset functors, Kripke polynomial functors form a relevant class of Set-functors in the theory of coalgebras. The main goal of this paper is to expand the theory of limits in categories of coalgebras of Kripke polynomial functors to the context of quantale-enriched categories. To assume the role of the powerset functor we consider “powerset-like” functors based on the Hausdorff V-category structure. As a starting point, we show that for a lifting of a Set-functor to a topological category X over Set that commutes with the forgetful functor, the corresponding category of coalgebras over X is topological over the category of coalgebras over Set and, therefore, it is “as complete” but cannot be “more complete”. Secondly, based on a Cantor-like argument, we observe that Hausdorff functors on categories of quantale-enriched categories do not admit a terminal coalgebra. Finally, in order to overcome these “negative” results, we combine quantale-enriched categories and topology à la Nachbin. Besides studying some basic properties of these categories, we investigate “powerset-like” functors which simultaneously encode the classical Hausdorff metric and Vietoris topology and show that the corresponding categories of coalgebras of “Kripke polynomial” functors are (co)complete. Keywords Coalgebra · Metric space · Compact space · Hausdorff metric · Vietoris topology
Communicated by Communicated by M. M. Clementino. This work is financed by the ERDF—European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation—COMPETE 2020 Programme and by National Funds through the Portuguese funding agency, FCT—Fundação para a Ciência e a Tecnologia, within Project POCI-01-0145-FEDER-030947, and Project UID/MAT/04106/2019 (CIDMA).
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Dirk Hofmann [email protected] Pedro Nora [email protected]
1
Department of Mathematics, Center for Research and Development in Mathematics and Applications, University of Aveiro, Aveiro, Portugal
2
HASLab INESC TEC - Institute for Systems and Computer Engineering, Technology and Science, University of Minho, Braga, Portugal
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D. Hofmann, P. Nora
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Strict Functorial Liftings . . . . . . . . . . . . . . . . . . . . . 3 Hausdorff Polynomial Functors on V-Cat . . . . . . . . . . . . 3.1 The Hausdorff Functor on V-Cat . . . . . . . . . . . . . . 3.2 Coalgebras of Hausdorff Polynomial Functors on V-Cat . . 4 Hausdorff Polynomial Functors on V-CatCH . . . . . . . . . . 4.1 Adding Topology . . . . . . . . . . . . . . . . . . . . . . 4.2 Coalgebras of Hausdorff Polynomial Functors on V-CatCH A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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