Abstract
This paper is concerned with final coalgebra representations of fractal sets. The background to our work includes Freyd’s Theorem: the unit interval is a final coalgebra of a certain endofunctor on the category of bipointed sets. Leinster’s far-ranging generalization of Freyd’s Theorem is also a central part of the discussion, but we do not directly build on his results. Our contributions are in two different directions. First, we demonstrate the connection of final coalgebras and initial algebras; this is an alternative development to one of his central contributions, working with resolutions.Second, we are interested in the metric space aspects of fractal sets. We work mainly with two examples: the unit interval [0,1] and the Sierpiński gasket \(\mathbb{S}\) as a subset of ℝ2.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adámek, J.: Free algebras and automata realizations in the language of categories. Comment. Math. Univ. Carolinae 14, 589–602 (1974)
Adámek, J.: On final coalgebras of continuous functors. Theoret. Comput. Sci. 294(1-2), 3–29 (2003)
Adámek, J., Milius, S., Moss, L.S.: Initial Algebras and Terminal Coalgebras: a Survey (2011) (unpublished ms.)
Aichholzer, O.: The path of a triangulation. In: Proc. 15th Ann. ACM Symp. Computational Geometry, Miami Beach, Florida, USA, pp. 14–23 (1999)
Barr, M.: Terminal coalgebras in well-founded set theory. Theoret. Comput. Sci. 114(2), 299–315 (1993); Additions and corrections in Theoret. Comput. Sci. 124 (1), 189–192 (1994)
Freyd, P.: Real coalgebra, post on the Categories mailing list (December 22, 1999), www.mta.ca/~cat-dist
Hasuo, I., Jacobs, B., Niqui, M.: Coalgebraic representation theory of fractals (Extended Abstract). In: Proc. Mathematical Foundations of Programming Semantics (MFPS XXVI), Electr. Notes Comp. Sci., vol. 265, pp. 351–368 (2010)
Hutchinson, J.: Fractals and self-similarity. Indiana Univ. Math. Journal 30(5), 713–747 (1981)
Lambek, J.: A Fixpoint Theorem for Complete Categories. Math. Z. 103, 151–161 (1968)
Leinster, T.: A general theory of self-similarity. Advances in Mathematics 226, 2935–3017 (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Bhattacharya, P., Moss, L.S., Ratnayake, J., Rose, R. (2014). Fractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra. In: van Breugel, F., Kashefi, E., Palamidessi, C., Rutten, J. (eds) Horizons of the Mind. A Tribute to Prakash Panangaden. Lecture Notes in Computer Science, vol 8464. Springer, Cham. https://doi.org/10.1007/978-3-319-06880-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-06880-0_7
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-06879-4
Online ISBN: 978-3-319-06880-0
eBook Packages: Computer ScienceComputer Science (R0)