Completely distributive enriched categories are not always continuous. (English) Zbl 1444.18007
Summary: In contrast to the fact that every completely distributive lattice is necessarily continuous in the sense of Scott, it is shown that complete distributivity of a category enriched over the closed category obtained by endowing the unit interval with a continuous t-norm does not imply its continuity in general. Necessary and sufficient conditions for the implication are presented.
MSC:
| 18B35 | Preorders, orders, domains and lattices (viewed as categories) |
| 18D20 | Enriched categories (over closed or monoidal categories) |
| 06D10 | Complete distributivity |
| 06F07 | Quantales |
Keywords:
enriched category; continuous t-norm; forward Cauchy weight; distributive law; completely distributive quantale-enriched category; continuous quantale-enriched categoryReferences:
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| [39] | Gabriella Böhm, Wigner Research Centre for Physics: bohm.gabriella (at) wigner.mta.hu Valeria de Paiva: Nuance Communications Inc: valeria.depaiva@gmail.com Richard Garner, Macquarie University: richard.garner@mq.edu.au Ezra Getzler, Northwestern University: getzler (at) northwestern(dot)edu |
| [40] | Kathryn Hess, Ecole Polytechnique Fédérale de Lausanne: kathryn.hess@epfl.ch Dirk Hofmann, Universidade de Aveiro: dirk@ua.pt Pieter Hofstra, Université d’ Ottawa: phofstra (at) uottawa.ca Anders Kock, University of Aarhus: kock@math.au.dk Joachim Kock, Universitat Autònoma de Barcelona: kock (at) mat.uab.cat Stephen Lack, Macquarie University: steve.lack@mq.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@buffalo.edu Tom Leinster, University of Edinburgh: Tom.Leinster@ed.ac.uk Matias Menni, Conicet and Universidad Nacional de La Plata, Argentina: matias.menni@gmail.com Ieke Moerdijk, Utrecht University: i.moerdijk@uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Kate Ponto, University of Kentucky: kate.ponto (at) uky.edu Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Giuseppe Rosolini, Università di Genova: rosolini@disi.unige.it Alex Simpson, University of Ljubljana: Alex.Simpson@fmf.uni-lj.si James Stasheff, University of North Carolina: jds@math.upenn.edu Ross Street, Macquarie University: ross.street@mq.edu.au Tim Van der Linden, Université catholique de Louvain: tim.vanderlinden@uclouvain.be |
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