# Hausdorff coalgebras

## TL;DR

This paper extends the theory of coalgebras for Kripke polynomial functors to quantale-enriched categories, exploring limits, non-existence of terminal coalgebras, and combining topology to achieve (co)completeness.

## Contribution

It introduces Hausdorff functors in quantale-enriched categories and demonstrates (co)completeness of their coalgebra categories, overcoming limitations of existing structures.

## Key findings

- Categories of coalgebras are topological over Set-based coalgebras.
- Hausdorff functors lack terminal coalgebras in quantale-enriched categories.
- Combining quantale-enriched categories with Nachbin topology yields (co)complete coalgebra categories.

## Abstract

As composites of constant, (co)product, identity, and powerset functors, Kripke polynomial functors form a relevant class of $\mathsf{Set}$-functors in the theory of coalgebras. The main goal of this paper is to expand the theory of limits in categories of coalgebras of Kripke polynomial functors to the context of quantale-enriched categories. To assume the role of the powerset functor we consider "powerset-like" functors based on the Hausdorff $\mathsf{V}$-category structure. As a starting point, we show that for a lifting of a $\mathsf{SET}$-functor to a topological category $\mathsf{X}$ over $\mathsf{Set}$ that commutes with the forgetful functor, the corresponding category of coalgebras over $\mathsf{X}$ is topological over the category of coalgebras over $\mathsf{Set}$ and, therefore, it is "as complete" but cannot be "more complete". Secondly, based on a Cantor-like argument, we observe that Hausdorff functors on categories of quantale-enriched categories do not admit a terminal coalgebra. Finally, in order to overcome these "negative" results, we combine quantale-enriched categories and topology \emph{\`a la} Nachbin. Besides studying some basic properties of these categories, we investigate "powerset-like" functors which simultaneously encode the classical Hausdorff metric and Vietoris topology and show that the corresponding categories of coalgebras of "Kripke polynomial" functors are (co)complete.