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Types and coalgebraic structure. (English) Zbl 1086.08002
Summary: We relate weak limit preservation properties of coalgebraic type functors \(F\) to structure-theoretic properties of the class \(\mathcal Set_F\) of all \(F\)-coalgebras. In particular, we give coalgebraic characterizations for the condition that \(F\) weakly preserves pullbacks, kernel pairs or preimages. We also describe regular monos and epis. In case that \(| F| \not= 1\) we show that \(F\) preserves preimages iff \(\mathcal{HS}(\mathcal K) = \mathcal{SH}(\mathcal K)\) for every class \(\mathcal K\) of \(F\)-coalgebras. The case \(| F(1)| = 1\) is left as an open problem.

MSC:
08A70 Applications of universal algebra in computer science
18A35 Categories admitting limits (complete categories), functors preserving limits, completions
18B20 Categories of machines, automata
68Q65 Abstract data types; algebraic specification
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