  Bisimilarity as an equivalence notion of systems has been central to process
theory. Due to the recent rise of interest in quantitative systems
(probabilistic, weighted, hybrid, etc.), bisimilarity has been extended in
various ways: notably, bisimulation metric between probabilistic systems. An
important feature of bisimilarity is its game-theoretic characterization, where
Spoiler and Duplicator play against each other; extension of bisimilarity games
to quantitative settings has been actively pursued too. In this paper, we
present a general framework that uniformly describes game characterizations of
bisimilarity-like notions. Our framework is formalized categorically using
fibrations and coalgebras. In particular, our characterization of bisimilarity
in terms of fibrational predicate transformers allows us to derive codensity
bisimilarity games: a general categorical game characterization of
bisimilarity. Our framework covers known bisimilarity-like notions (such as
bisimulation metric) as well as new ones (including what we call bisimulation
topology).
