Generalised powerlocales via relation lifting

This paper introduces an endofunctor VT on the category of frames that is parametrised by an endofunctor T on the category Set that satisfies certain constraints. This generalises Johnstone's construction of the Vietoris powerlocale in the sense that his construction is obtained by taking for T the finite covariant power set functor. Our construction of the T-powerlocale VT out of a frame is based on ideas from coalgebraic logic and makes explicit the connection between the Vietoris construction and Moss's coalgebraic cover modality.We show how to extend certain natural transformations between set functors to natural transformations between T-powerlocale functors. Finally, we prove that the operation VT preserves some properties of frames, such as regularity, zero-dimensionality and the combination of zero-dimensionality and compactness.

Initial algebras and terminal coalgebras in many-sorted sets

We prove that the iterative construction of initial algebras converges for endofunctors F of many-sorted sets whenever F has an initial algebra. In the case of one-sorted sets, the convergence takes n steps where n is either an infinite regular cardinal or is at most 3. Dually, the existence of a many-sorted terminal coalgebra implies that the iterative construction of a terminal coalgebra converges. Moreover, every endofunctor with a fixed-point pair larger than the number of sorts is proved to have a terminal coalgebra. As demonstrated by James Worell, the number of steps here need not be a cardinal even in the case of a single sort: it is ω + ω for the finite power-set functor. The above results do not hold for related categories, such as graphs: we present non-constructive initial algebras and terminal coalgebras.