  The category of presheaves on a (small) category is a suitable semantic
universe to study behaviour of various dynamical systems. In particular,
presheaves can be used to record the executions of a system and their morphisms
correspond to simulation maps for various kinds of state-based systems. In this
paper, we introduce a notion of bisimulation maps between presheaves (or
executions) to capture well known behavioural equivalences in an abstract way.
We demonstrate the versatility of this framework by working out the
characterisations for standard bisimulation, $\forall$-fair bisimulation, and
branching bisimulation.
