  In various situations in Floer theory, one extracts homological invariants
from "Morse-Bott" data in which the "critical set" is a union of manifolds, and
the moduli spaces of "flow lines" have evaluation maps taking values in the
critical set. This requires a mix of analytic arguments (establishing
properties of the moduli spaces and evaluation maps) and formal arguments
(defining or computing invariants from the analytic data). The goal of this
paper is to isolate the formal arguments, in the case when the critical set is
a union of circles. Namely, we state axioms for moduli spaces and evaluation
maps (encoding a minimal amount of analytical information that one needs to
verify in any given Floer-theoretic situation), and using these axioms we
define homological invariants. More precisely, we define a (almost) category of
"Morse-Bott systems". We construct a "cascade homology" functor on this
category, based on ideas of Bourgeois and Frauenfelder, which is "homotopy
invariant". This machinery is used in our work on cylindrical contact homology.
