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1 results

  • 11 - Sufficient Conditions for 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 363-397

  • Summary

    This chapter presents a number of sufficient conditions to guarantee that an endofunctor has an initial algebra or a terminal coalgebra. We generalize Kawahara and Mori’s notion of a bounded set functor and prove that for a cocomplete and co-well-powered category with a terminal object, every endofunctor bounded by a generating set has a terminal coalgebra. We use this to show that every accessible endofunctor on a locally presentable category has an initial algebra and a terminal coalgebra. We introduce pre-accessible functors and prove that on a cocomplete and co-well-powered category, the initial-algebra chain of a pre-accessible functor converges, and so the initial algebra exists. If the base category is locally presentable and the functor preserves monomorphisms, then the terminal coalgebra exists.