Probability measure monad on the category of ultrametric spaces

Authors

  • O.B. Hubal Lviv National University
  • M.M. Zarichnyi Lviv National University

DOI:

https://doi.org/10.4995/agt.2008.1803

Keywords:

Probability measure, Ultrametric space, Monad, Kleisli category

Abstract

The set of all probability measures with compact support on an ultrametric space can be endowed with a natural ultrametric. We show that the functor of probability measures with finite supports (respectively compact supports) forms a monad in the category of ultrametric spaces (respectively complete ultrametric spaces) and nonexpanding maps. It is also proven that the G-symmetric power functor has an extension onto the Kleisli category of the probability measure monad.

Downloads

Download data is not yet available.

References

J. W. de Bakker and J. I. Zucker, Processes and the denotational semantic of concurrency, Information and Control 54 (1982), 70–120.

http://dx.doi.org/10.1016/S0019-9958(82)91250-5

M. Barr, Ch.Wells, Toposes, triples and theories. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 278. Springer-Verlag, New York, 1985. xiii+345 pp.

G. David, S. Semmes, Fractured Fractals and Broken Dreams: Self-Similar Geometry through Metric and Measure. - Oxford Lecture Ser. Math. Appl. 7, Oxford University Press, 1997.

J. I. den Hartog and E. P. de Vink, Building metric structures with the Meas functor, Duke Math. J. 34 (1967), 255–271; errata 813–814.

J. Heinonen, Lectures on analysis on metric spaces. Springer-Verlag, New York, 2001.

http://dx.doi.org/10.1007/978-1-4613-0131-8

J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30, no. 5 (1981), 713–747.

http://dx.doi.org/10.1512/iumj.1981.30.30055

T. ´ Swirszcz, Monadic functors and convexity. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 22 (1974), 39–42.

E. P. de Vink and J. J. M. M. Rutten, Bisimulation for probabilistic transition systems: a coalgebraic approach, Theoretical Computer Science 221, no. 1/2 (1999), 271–293.

http://dx.doi.org/10.1016/S0304-3975(99)00035-3

M. M. Zarichny˘ı, Characterization of functors of G-symmetric degree and extensions of functors to the Kleisli categories. (Russian) Mat. Zametki 52, no. 5 (1992), 42–48, 141; translation in Math. Notes 52, no. 5-6 (1992), 1107–1111 (1993).

Downloads

How to Cite

[1]
O. Hubal and M. Zarichnyi, “Probability measure monad on the category of ultrametric spaces”, Appl. Gen. Topol., vol. 9, no. 2, pp. 229–237, Oct. 2008.

Issue

Section

Regular Articles