Factorization systems for restriction categories. (English) Zbl 07902415
Latent fibrations [R. Cockett et al., Theory Appl. Categ. 36, 423–491 (2021; Zbl 1467.18009)] are the analogue of fibrations for restriction categories [J. R. B. Cockett and S. Lack, Theor. Comput. Sci. 270, No. 1–2, 223–259 (2002; Zbl 0988.18003); Theor. Comput. Sci. 294, No. 1–2, 61–102 (2003; Zbl 1023.18005); Math. Struct. Comput. Sci. 17, No. 4, 775–817 (2007; Zbl 1123.18003)]. The papers [J. R. B. Cockett et al., Theory Appl. Categ. 26, 412–452 (2012; Zbl 1252.18003); Theory Appl. Categ. 26, 453–500 (2012; Zbl 1252.18004)] provided a source of examples of restriction categories with a latent factorization. The principal objective in this paper is to get an analogous result for latent fibrations to that of ordinary fibrations. As restriction categories are not self-dual, the notion of orthogonality for restriction categories takes a correspondlingly non-self-dual character. The main result of the paper (Theorem 4.5) claims that to have an M-category with an M-stable factorrization system is precisely to have a split restriction category with a latent factorization.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18B10 | Categories of spans/cospans, relations, or partial maps |
| 18A32 | Factorization systems, substructures, quotient structures, congruences, amalgams |
Citations:
Zbl 1467.18009; Zbl 0988.18003; Zbl 1023.18005; Zbl 1123.18003; Zbl 1252.18003; Zbl 1252.18004References:
| [1] | Cassidy, C., Hebert, M., Kelly, G.M. Reflexive subcategories, localisations and factori-sation systems. Journal of the Australian Math. Soc.(Series A) 38 (1985) · Zbl 0573.18002 |
| [2] | Cockett J.R.B.,and Lack S., Restriction Categories I: Categories of partial maps. The-oretical Computer Science, Vol 270 (2002) 223-259. · Zbl 0988.18003 |
| [3] | Cockett J.R.B. , Guo X., and Hofstra P.J.W Range Categories I: General Theory Theory and Applications of Categories, Vol. 26, No. 17 (2012) 412-452. · Zbl 1252.18003 |
| [4] | Cockett J.R.B., Guo X., and Hofstra P.J.W. Range Categories II: Towards Regularity Theory and Applications of Categories, Vol. 26, No. 18 (2012) 454-500. |
| [5] | Cockett J.R.B., Cruttwell G.S.H., Gallagher J.D., and Pronk D.A. Latent Fibrations: Fibrations for Categories of Partial Maps Theory and Applications of Categories, Vol. 36, 2021, No. 16, pp 423-491. · Zbl 1467.18009 |
| [6] | Grandis M. Manifolds and local structures: a general theory. World Scientific (2021). · Zbl 1487.58001 |
| [7] | Herrlich H., Salicrup G.,and Strecker G.E. Factorizations, denseness, separation, and relatively compact objects. Topology and its Applications 27 (1987) 157-169. · Zbl 0629.18003 |
| [8] | Hofmann D., Seal G.J., and Tholen W., eds. Monoidal Topology: A Categorical Ap-proach to Order, Metric, and Topology. Cambridge University Press (2014). · Zbl 1297.18001 |
| [9] | P.T.Johnstone P.T. Sketches of an elephant. A compendium of Topos theory. 2 vols. Oxford University Press(2002). · Zbl 1071.18001 |
| [10] | THEORY AND APPLICATIONS OF CATEGORIES will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods. Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication. Subscription information Individual subscribers receive abstracts of articles by e-mail as they are published. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. Full text of the journal is freely available at http://www.tac.mta.ca/tac/. |
| [11] | Information for authors L A T E X2e is required. Articles may be submitted in PDF by email directly to a Transmitting Editor following the author instructions at http://www.tac.mta.ca/tac/authinfo.html. Managing editor. Geoff Cruttwell, Mount Allison University: gcruttwell@mta.ca T E Xnical editor. Michael Barr, McGill University: michael.barr@mcgill.ca |
| [12] | Assistant T E X editor. Gavin Seal, Ecole Polytechnique Fédérale de Lausanne: gavin seal@fastmail.fm Transmitting editors. |
| [13] | Clemens Berger, Université de Nice-Sophia Antipolis: cberger@math.unice.fr Julie Bergner, University of Virginia: jeb2md (at) virginia.edu Richard Blute, Université d’ Ottawa: rblute@uottawa.ca John Bourke, Masaryk University: bourkej@math.muni.cz Maria Manuel Clementino, Universidade de Coimbra: mmc@mat.uc.pt Valeria de Paiva, Topos Institute: valeria.depaiva@gmail.com Richard Garner, Macquarie University: richard.garner@mq.edu.au |
| [14] | Ezra Getzler, Northwestern University: getzler (at) northwestern(dot)edu Rune Haugseng, Norwegian University of Science and Technology: rune.haugseng@ntnu.no Dirk Hofmann, Universidade de Aveiro: dirk@ua.pt |
| [15] | Joachim Kock, Universitat Autònoma de Barcelona: Joachim.Kock (at) uab.cat Stephen Lack, Macquarie University: steve.lack@mq.edu.au Tom Leinster, University of Edinburgh: Tom.Leinster@ed.ac.uk |
| [16] | Sandra Mantovani, Università degli Studi di Milano: sandra.mantovani@unimi.it Matias Menni, Conicet and Universidad Nacional de La Plata, Argentina: matias.menni@gmail.com Giuseppe Metere, Università degli Studi di Palermo: giuseppe.metere (at) unipa.it Kate Ponto, University of Kentucky: kate.ponto (at) uky.edu Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca Jiri Rosický, Masaryk University: rosicky@math.muni.cz Giuseppe Rosolini, Università di Genova: rosolini@unige.it Michael Shulman, University of San Diego: shulman@sandiego.edu |
| [17] | Alex Simpson, University of Ljubljana: Alex.Simpson@fmf.uni-lj.si James Stasheff, University of North Carolina: jds@math.upenn.edu Tim Van der Linden, Université catholique de Louvain: tim.vanderlinden@uclouvain.be Christina Vasilakopoulou, National Technical University of Athens: cvasilak@math.ntua.gr |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
