Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Tbilisi Mathematical Journal

Editor-in-Chief: Inassaridze, Hvedri

2 Issues per year


Mathematical Citation Quotient (MCQ) 2016: 0.14

Open Access
Online
ISSN
1512-0139
See all formats and pricing
More options …

Real sets

George Janelidze / Ross Street
Published Online: 2017-09-20 | DOI: https://doi.org/10.1515/tmj-2017-0101

Abstract

After reviewing a universal characterization of the extended positive real numbers published by Denis Higgs in 1978, we define a category which provides an answer to the questions: what is a set with half an element? ∙ what is a set with π elements? The category of these extended positive real sets is equipped with a countable tensor product. We develop somewhat the theory of categories with countable tensors; we call the commutative such categories series monoidal and conclude by only briefly mentioning the non-commutative possibility called ω-monoidal. We include some remarks on sets having cardinalities in [-∞;∞].

MSC 2010: 18D10; 18D20; 20M14; 28A20

Keywords: Commutative monoid; biproduct; direct sum; abstract addition; magnitude module; series monoidal category

References

  • [1] Aristotle, Physics 6(9) (350 B.C.E) 239b10.Google Scholar

  • [2] Jean Bénabou, Introduction to bicategories, Lecture Notes in Mathematics 47 (Springer-Verlag, 1967) 1-77.Google Scholar

  • [3] Robert Blackwell, G. Max Kelly and A. John Power, Two-dimensional monad theory, J. Pure Appl. Algebra 59 (1989) 1-41.Google Scholar

  • [4] Brian Day, *-Autonomous categories in quantum theory, www.arxiv.org/abs/math/0605037 5 pp.Google Scholar

  • [5] Brian Day and Ross Street, Monoidal bicategories and Hopf algebroids, Advances in Math. 129 (1997) 99-157.Google Scholar

  • [6] Samuel Eilenberg and G.M. Kelly, Closed categories, Proceedings of the Conference on Categorical Algebra (La Jolla, 1965), (Springer-Verlag,1966) 421-562.Google Scholar

  • [7] J. Fillmore, D. Pumplün and H. Röhrl, On N-summations, I, Applied Categorical Structures 10 (2002) 291-315.Google Scholar

  • [8] Peter Freyd, Algebraic real analysis, Theory and Applications of Categories, 20(10) (2008) 215-306.Google Scholar

  • [9] Denis Higgs, A universal characterization of [0;1], Nederl. Akad. Wetensch. Indag. Math. 40(4) (1978) 448-455.Google Scholar

  • [10] Denis Higgs, Axiomatic infinite sums - an algebraic approach to integration theory, Contemp. Math. 2 (Amer. Math. Soc., 1980) 205-212.Google Scholar

  • [11] E.V. Huntingdon, A complete set of postulates for the theory of absolute continuous magnitude, Transactions Amer. Math. Soc. 3 (1902) 264-279.CrossrefGoogle Scholar

  • [12] André Joyal and Ross Street, The geometry of tensor calculus, I, Advances in Mathematics 88 (1991) pp. 55-112.Google Scholar

  • [13] André Joyal, Ross Street and Dominic Verity Traced monoidal categories, Mathematical Proceedings of the Cambridge Philosophical Society 119(3) (1996) 425-446.Google Scholar

  • [14] G.M. Kelly, Many-variable functorial calculus I, Lecture Notes in Math. 281 (Springer-Verlag, 1972) 66-105.Google Scholar

  • [15] G.M. Kelly, Doctrinal adjunction, Lecture Notes in Mathematics 420 (Springer-Verlag, 1974) 257-280.Google Scholar

  • [16] G.M. Kelly, Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series 64 (Cambridge University Press, Cambridge, 1982).Google Scholar

  • [17] G.M. Kelly and Ross Street, Review of the elements of 2-categories, Lecture Notes in Mathematics 420 (Springer-Verlag, 1974) 75-103.Google Scholar

  • [18] Anders Kock, Closed categories generated by commutative monads, J. Australian Math. Soc. 12 (1971) 405-424.CrossrefGoogle Scholar

  • [19] F. W. Lawvere, Metric spaces, generalized logic and closed categories, Reprints in Theory and Applications of Categories 1 (2002) pp.1-37; originally published as: Rendiconti del Seminario Matematico e Fisico di Milano 53 (1973) 135-166.Google Scholar

  • [20] Saunders Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5 (Springer-Verlag, 1971).Google Scholar

  • [21] Chi-Keung Ng, On genuine infinite algebraic tensor products, Revista Matemática Iberoamericana, 29(1) (2013) 329-356.Google Scholar

  • [22] Stephen H. Schanuel, Negative sets have Euler characteristic and dimension, Lecture Notes in Mathematics 1488 (Springer, 1991) 379-385.Google Scholar

  • [23] Zbigniew Semadeni, Monads and their Eilenberg-Moore algebras in functional analysis, Queen's Papers in Pure and Applied Mathematics 33 (Queen's University, Kingston, Ont., 1973) iii+98 pp.Google Scholar

  • [24] Ross Street, Fibrations in bicategories, Cahiers de topologie et géométrie différentielle 21 (1980) 111-160.Google Scholar

  • [25] Ross Street, An efficient construction of the real numbers, Gazette Australian Math. Soc. 12 (1985) 57-58; also see http://maths.mq.edu.au/~street/EffR.pdf.Google Scholar

  • [26] Ross Street, Skew-closed categories, Journal of Pure and Applied Algebra 217(6) (2013) 973-988.Google Scholar

  • [27] Ross Street, Weighted tensor products of Joyal species, graphs, and charades, SIGMA 12(005) (2016) 20pp.Google Scholar

  • [28] Alfred Tarski, Cardinal algebras (New York, Oxford University Press, 1949).Google Scholar

  • [29] Alfred Tarski, Ordinal algebras (North-Holland, Amsterdam, 1956).Google Scholar

  • [30] Stephen T. Welstead, Infinite products in a Banach algebra, Journal of Math. Analysis and Applications 105 (1985) 523-532.Google Scholar

About the article

Received: 2017-05-15

Accepted: 2017-09-07

Published Online: 2017-09-20

Published in Print: 2017-09-26


Citation Information: Tbilisi Mathematical Journal, ISSN (Online) 1512-0139, DOI: https://doi.org/10.1515/tmj-2017-0101.

Export Citation

© 2017. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (0)

Please log in or register to comment.
Log in