Real sets

Open access

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 [-∞;∞].

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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.

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

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

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

[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.

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

[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.

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

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

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

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

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