  In this paper, we investigate diagrams, namely functors from any small
category to a fixed category, and more particularly, their bisimilarity.
Initially defined using the theory of open maps of Joyal et al., we prove
several equivalent characterizations: it is equivalent to the existence of a
relation, similar to history-preserving bisimulations for event structures and
it has a logical characterization similar to the Hennessy-Milner theorem. We
then prove that we capture many different known bismilarities, by considering
the category of executions and extensions of executions, and by forming the
functor that maps every execution to the information of interest for the
problem at hand. We then look at the particular case of finitary diagrams with
values in real or rational vector spaces. We prove that checking bisimilarity
and satisfiability of a positive formula by a diagram are both decidable by
reducing to a problem of existence of invertible matrices with linear
conditions, which in turn reduces to the existential theory of the reals.
