A note on Frobenius-Eilenberg-Moore objects in dagger 2-categories. (English) Zbl 1473.18002
Dagger Frobenius monads and categories of Frobenius-Eilenberg-Moore algebras for such monads were first considered in [C. Heunen and M. Karvonen, Theory Appl. Categ. 31, 1016–1043 (2016; Zbl 1378.18003); Electron. Notes Theor. Comput. Sci. 319, 217–237 (2015; Zbl 1351.68100)], where it was shown that they include the important examaple of quantum measurements. This paper pursues a formal theory of dagger Frobenius monads in the spirit of [R. Street, J. Pure Appl. Algebra 2, 149–168 (1972; Zbl 0241.18003); S. Lack and R. Street, J. Pure Appl. Algebra 175, No. 1–3, 243–265 (2002; Zbl 1019.18002)]. It is shown that the free completion of a \(2\)-category under Eilenberg-Moore objects extends to the dagger context, provided one is ready to work with those dagger Frobenius monads for which the endofunctor suitably commutes with the unit. Defining dagger lax functors and dagger lax-limits of such functors, the author demonstrates that Frobenius-Eilenberg-Moore objects are examples of such limits.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18A35 | Categories admitting limits (complete categories), functors preserving limits, completions |
| 18A40 | Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) |
| 18C15 | Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads |
| 18C20 | Eilenberg-Moore and Kleisli constructions for monads |
| 18D70 | Formal category theory |
| 18N10 | 2-categories, bicategories, double categories |
| 18N15 | 2-dimensional monad theory |
References:
| [1] | C. Heunen and M. Karvonen. Monads on dagger categories. Theory and Applications of Categories, 31(35):1016-1043, 2016 · Zbl 1378.18003 |
| [2] | C. Heunen and M. Karvonen. Reversible Monadic Computing. Electronic Notes in Theo-retical Computer Science, 319:217-237, 2015 · Zbl 1342.68016 |
| [3] | C. Heunen and J. Vicary, Categories for Quantum Theory: An Introduction. Oxford Uni-versity Press, Oxford Graduate Texts in Mathematics, 2019. · Zbl 1436.81004 |
| [4] | M. Karvonen, PhD Thesis: The Way of the Dagger. University of Edinburgh, 2019. |
| [5] | G. M. Kelly. Basic Concepts of Enriched Category Theory. Reprints in Theory and Ap-plications of Categories, 1(10):1-136, 2005 · Zbl 1086.18001 |
| [6] | J. Kock, Frobenius Algebras and 2-D Topological Quantum Field Theories. Cambridge University Press, London Mathematical Society Student Texts, 2003. |
| [7] | A. Lauda. Frobenius algebras and ambidextrous adjunctions. Theory and Applications of Categories, 16(4):84-122, 2006. · Zbl 1092.18003 |
| [8] | R. Street, Frobenius monads and pseudomonoids. Journal of Mathematical Physics, 45(10):3930-3948, 2004 · Zbl 1071.18006 |
| [9] | R. Street, Limits indexed by category-valued 2-functors. Journal of Pure and Applied Algebra, 8(2):149-181, 1976. · Zbl 0335.18005 |
| [10] | R. Street, The formal theory of monads. Journal of Pure and Applied Algebra, 2(2):149-168, 1972. · Zbl 0241.18003 |
| [11] | S. Lack and R. Street, The formal theory of monads II. Journal of Pure and Applied Algebra, 175(1):243-265, 2002. · Zbl 1019.18002 |
| [12] | Department of Mathematics and Applied Mathematics, University of Cape Town Rondebosch 7701 Email: PKLROW001@myuct.ac.za |
| [13] | Gabriella Böhm, Wigner Research Centre for Physics: bohm.gabriella (at) wigner.mta.hu Maria Manuel Clementino, Universidade de Coimbra: mmc.mat.uc.pt 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 |
| [14] | 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 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 |
| [15] | Kate Ponto, University of Kentucky: kate.ponto (at) uky.edu Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca Jiří Rosický, Masaryk University: rosicky@math.muni.cz Giuseppe Rosolini, Università di Genova: rosolini@disi.unige.it Michael Shulman, University of San Diego: shulman@sandiego.edu Alex Simpson, University of Ljubljana: Alex.Simpson@fmf.uni-lj.si James Stasheff, University of North Carolina: jds@math.upenn.edu |
| [16] | Tim Van der Linden, Université catholique de Louvain: tim.vanderlinden@uclouvain.be |
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