  There are two rather distinct approaches to Morse theory nowadays: smooth and
discrete. We propose to study a real valued function by assembling all
associated sections in a topological category. From this point of view, Reeb
functions on stratified spaces are introduced, including both smooth and
combinatorial examples. As a consequence of the simplicial approach taken, the
theory comes with a spectral sequence for computing (generalized) homology. We
also model the homotopy type of Reeb graphs/ topological Reeb spaces as
simplicial sets, which are combinatorial in nature, as opposed to the typical
description in terms of quotient spaces.
