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1 results

  • 13 - Interaction between Initial Algebras and Terminal Coalgebras

  • Jiří Adámek, Czech Technical University in Prague, Stefan Milius, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany, Lawrence S. Moss, Indiana University, Bloomington
  • Book:
    Initial Algebras and Terminal Coalgebras
    Published online:
    30 January 2025
    Print publication:
    06 February 2025, pp 461-476

  • Summary

    The theme of this chapter is the relation between the initial algebra for a set functor and the terminal coalgebra, assuming that both exist and that the endofunctor is non-trivial. We introduced a notion called pre-continuity. Pre-continuous set functors generalize finitary and continuous set functors. For such functors, the initial algebra and the terminal coalgebra have the same Cauchy completion and the same ideal completion: the $\omega$-iteration of the terminal-coalgebra chain. It follows that for a non-trivial continuous set functor, the terminal coalgebra is the Cauchy completion of the initial algebra.