  Categorical studies of recursive data structures and their associated
reasoning principles have mostly focused on two extremes: initial algebras and
induction, and final coalgebras and coinduction. In this paper we study their
in-betweens. We formalize notions of alternating fixed points of functors using
constructions that are similar to that of free monads. We find their use in
categorical modeling of accepting run trees under the Buechi and parity
acceptance condition. This modeling abstracts away from states of an automaton;
it can thus be thought of as the "behaviors" of systems with the Buechi or
parity conditions, in a way that follows the tradition of coalgebraic modeling
of system behaviors.
