Models of Non-Well-Founded Sets via an Indexed Final Coalgebra Theorem

The paper uses the formalism of indexed categories to recover the proof of a standard final coalgebra theorem, thus showing existence of final coalgebras for a special class of functors on categories with finite limits and colimits. As an instance of this result, we build the final coalgebra for the powerclass functor, in the context of a Heyting pretopos with a class of small maps. This is then proved to provide a model for various non-well-founded set theories, depending on the chosen axiomatisation for the class of small maps.