Abstract
We extend the work of Schanuel, Lawvere, Blass and Gates in Objective Number Theory by proving that, for any \({L(X) \in \mathbb {N}[X]}\), the rig \({\mathbb {N}[X]/(X = L(X))}\) is the Burnside rig of a prextensive category.
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Bénabou, J.: Introduction to Bicategories. In: Lecture Notes in Mathematics, Volume 47, pp 1–77. Springer-Verlag, New York, Berlin (1967)
Betti, R., Galuzzi, M.: Categorie normate. Boll. Un. Mat. Ital.(4) 11(1), 66–75 (1975)
Blass, A.: Seven trees in one. J. Pure Appl. Algebra 103(1), 1–21 (1995)
Borceux, F.: Handbook of categorical algebra 1 volume 50 of Encyclopedia of mathematics and its applications. Cambridge University Press (1994)
Carboni, A., Lack, S., Walters, R.F.C.: Introduction to extensive and distributive categories. Journal of Pure and Applied Algebra 84, 145–158 (1993)
Diener, K.-H.: On the predecessor relation in abstract algebras. Math. Logic Quart. 39(4), 492–514 (1993)
Gabriel, P., Zisman, M.: Calculus of Fractions and Homotopy Theory. Ergebnisse Der Mathematik Und Ihrer Grenzgebiete Band, vol. 35. Springer-Verlag, Berlin-Heidelberg-New York (1967)
Gates, R.: On Extensive and Distributive Categories. PhD Thesis, School of Mathematics and Statistics. University of Sydney, Australia (1997)
Johnstone, P.T.: Topos theory. Academic Press (1977)
Johnstone, P. T.: Sketches of an Elephant: a Topos Theory Compendium, Volume 43-44 of Oxford Logic Guides. The Clarendon Press Oxford University Press, New York (2002)
Lawvere, F.W.: Some Thoughts on the Future of Category Theory. In: Proceedings of Category Theory 1990, Como, Italy, volume 1488 of Lecture notes in mathematics, pp 1–13, Springer-Verlag (1991)
Lawvere, F.W.: Metric spaces, generalized logic, and closed categories[Rend. Sem. Mat. Fis. Milano 43 (1973), 135–166 (1974)]. Repr. Theory Appl. Categ., (1):1–37. With an author commentary: Enriched categories in the logic of geometry and analysis (2002)
Lawvere, F.W.: Functorial semantics of algebraic theories and some algebraic problems in the context of functorial semantics of algebraic theories. Repr. Theory Appl. Categ. 2004(5), 1–121 (2004)
Lawvere, F.W.: Core varieties, extensivity, and rig geometry. Theory Appl. Categ. 20(14), 497–503 (2008)
Menni, M.: Bimonadicity and the explicit basis property. Theory Appl. Categ. 26, 554–581 (2012)
Schanuel, S.H.: Negative sets have Euler characteristic and dimension. Category theory. Proc. Int. Conf., Como/Italy 1990, Lect. Notes Math. 1488, 379–385 (1991)
Schanuel, S.H.: Objective number theory and the retract chain condition. J. Pure Appl. Algebra 154(1-3), 295–298 (2000). Category theory and its applications (Montreal, QC, 1997)
Schanuel, S.H.: Transcendence in objective number theory. In Categorical studies in Italy, Perugia, Italy, 1997. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 43–48, 64 (2000)
Szawiel, S., Zawadowski, M.: Theories of analytic monads. Math. Structures Comput. Sci. 24(6) (2014)
Acknowledgments
I would like to thank F. W. Lawvere for posing the original problem that led to this paper and for his encouragement since then. I also want to thank M. Zawadowski for his advice on analytic theories. The referee made several useful suggestions which simplified the original exposition.
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Menni, M. Every Rig with a One-Variable Fixed Point Presentation is the Burnside Rig of a Prextensive Category. Appl Categor Struct 25, 663–707 (2017). https://doi.org/10.1007/s10485-016-9475-6
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DOI: https://doi.org/10.1007/s10485-016-9475-6