Relating Coalgebraic Notions of Bisimulation

A labelled transition system can be understood as a coalgebra for a particular endofunctor on the category of sets. Generalizing, we are led to consider coalgebras for arbitrary endofunctors on arbitrary categories. Bisimulation is a crucial notion in the theory of labelled transition systems. We identify four definitions of bisimulation on general coalgebras. The definitions all specialize to the same notion for the special case of labelled transition systems. We investigate general conditions under which the four notions coincide. As an extended example, we consider the semantics of name-passing process calculi (such as the pi-calculus), and present a new coalgebraic model for name-passing calculi.