Fractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra

  • Prasit Bhattacharya
  • Lawrence S. Moss
  • Jayampathy Ratnayake
  • Robert Rose
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8464)


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.


Unit Interval Isometric Embedding Initial Object Connection Point Algebra Morphism 
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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Prasit Bhattacharya
    • 1
  • Lawrence S. Moss
    • 1
  • Jayampathy Ratnayake
    • 1
  • Robert Rose
    • 1
  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA

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