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Final coalgebras are ideal completions of initial algebras. (English) Zbl 1003.18009

Let \(\mathcal K\) be a category with an initial object \(0\), a terminal object \(1\), limits of \(\omega^{\text{op}}\)-chains and colimits of \(\omega\)- chains. An endofunctor \(F\) of \(\mathcal K\) is called \(\omega\)-continuous if \(F\) preserves limits of \(\omega^{\text{op}}\)-chains and is called grounded if there exists a \(\mathcal K\)-morphism from \(1\) into \(F0\). It is well known that if \(F\) is a \(\omega\)-continuous set functor then a final \(F\)-coalgebra exists and there exists a natural ordering on it. The paper proves that if \(F\) is an \(\omega\)-continuous grounded set endofunctor then an initial \(F\)-algebra exists and a final \(F\)-coalgebra is an ideal completion of an initial \(F\)-algebra. This result is generalized for a locally finitely presentable category \(\mathcal K\) such that the initial \(\mathcal K\)-object has no proper quotient. The obtained results are illustrated in many examples.

MSC:

18C50 Categorical semantics of formal languages
18A35 Categories admitting limits (complete categories), functors preserving limits, completions
18C10 Theories (e.g., algebraic theories), structure, and semantics
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
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