We introduce and study bisimulations for coalgebras on Stone spaces [14]. Our notion of bisimulation is sound and complete for behavioural equivalence, and generalizes Vietoris bisimulations [3]. The main result of our paper is that bisimulation for a Stone coalgebra is the topological closure of bisimulation for the underlying Set coalgebra.

In this article, we investigate the functors from modules to modules that occur as the summands of tensor powers and the functors from modules to Hopf algebras that occur as natural coalgebra summands of tensor algebras. The main results provide some explicit natural coalgebra summands of tensor algebras. As a consequence, we obtain some decompositions of Lie powers over the general linear groups.

Let H be a quasi-Hopf algebra. We show that any H-bimodule coalgebra C for which there exists an H-bimodule coalgebra morphism ν : C → H is isomorphic to what we will call a smash product coalgebra. To this end, we use an explicit monoidal equivalence between the category of two-sided two-cosided Hopf modules over H and the category of left Yetter-Drinfeld modules over H. This categorical metho...

By a coalgebra over the commutative ring K or a K-coalgebra, we understand a cocommutative, coassociative K-coalgebra with counit. More explicitly we mean a K-module C equipped with maps &C+C&C, subject to the requirement (where we write @ for OK) that the following diagrams commute: CLC@C where the vertical arrow is the one which switches the factors; CAC@C 1 CC38 C@C e@'CtC@C@C, where we have...

A long and fertile tradition in theoretical computer science, going back to the work of Büchi and Rabin, links the field of (fixpoint) logic to that of automata theory. In particular, automata operating on potentially infinite structures such as streams, trees, graphs or transition systems, provide an invaluable tool for the specification and verification of the ongoing behavior of systems. An ...

Coalgebra is used to generalize notions and techniques from concurrency theory, in order to apply them to problems concerning the supervisory control of discrete event systems. The main ingredients of this approach are the characterization of controllability in terms of (a variant of) the notion of bisimulation, and the observation that the family of (partial) languages carries a final coalgebr...

Rough notes for the participants of the summer school affiliated to CTCS'04 held in Copen-hagen, 9-11 August 2004.