On quantale-enriched categories the Hausdorff functor has no terminal coalgebra, but on enriched compact Hausdorff spaces it preserves codirected limits, making categories of Hausdorff polynomial coalgebras complete.
Convergence and quantale-enriched categories
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abstract
Generalising Nachbin's theory of "topology and order", in this paper we continue the study of quantale-enriched categories equipped with a compact Hausdorff topology. We compare these $\mathcal{V}$-categorical compact Hausdorff spaces with ultrafilter-quantale-enriched categories, and show that the presence of a compact Hausdorff topology guarantees Cauchy completeness and (suitably defined) codirected completeness of the underlying quantale enriched category.
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math.CT 1years
2019 1verdicts
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Hausdorff coalgebras
On quantale-enriched categories the Hausdorff functor has no terminal coalgebra, but on enriched compact Hausdorff spaces it preserves codirected limits, making categories of Hausdorff polynomial coalgebras complete.