Classical Morse homology recovers the ordinary homology of a closed manifold using the data of critical points and gradient flows. In the 1990s, Cohen-Jones-Segal proposed a categorification of this data, called the "flow category", with an eye towards developing homotopical refinements of Floer homology, which marked the starting point of "Floer homotopy theory". We will show that under mild assumptions on the topology of moduli spaces, the classifying space of the flow category is homotopy equivalent to the underlying manifold, thereby addressing a gap in a preprint of Cohen-Jones-Segal. We also show that the assumption we put is crucial by describing a Morse function and metric on S^2xS^1 whose flow category fails to recover the homotopy type. This is based on joint work with Maxine Calle.