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.
Reviewer: Václav Koubek (Praha)
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.) |