  In this paper, we construct cochain complexes generated by the cohomology of
critical manifolds in the abstract setup of flow categories for Morse-Bott
theories under minimum transversality assumptions. We discuss the relations
between different constructions of Morse-Bott theories. In particular, we
explain how homological perturbation theory is used in Morse-Bott theories, and
both our construction and the cascades construction can be interpreted as
applications of homological perturbations. In the presence of group actions, we
construct cochain complexes for the equivariant theory. Expected properties
like the independence of approximations of classifying spaces and the existence
of the action spectral sequence are proven. We carry out our construction for
Morse-Bott functions on closed manifolds and prove it recovers the regular
cohomology. We outline the project of combining our construction with the
polyfold theory.
