Forman introduced discrete Morse theory on finite regular cell complexes. Minian introduced a version of discrete Morse theory for posets which generalizes Forman's original Morse theory https://arxiv.org/abs/1007.1930. So I thought the following problem:
Is there a version of discrete Morse theory for acyclic categories which generalizes Minian's Morse theory?
A desire theorem (It is imaginary and is not logical and currect.) is that there is a "function (functor)" from an acyclic category such that the classifying space of the given acyclic category is homotopy equivalent to a CW complex such that the number of cells = the number of "critical objects".
An acyclic category is a small category in which only the identity morphisms have inverses and any morphism from an object to itself is an identity morphism. A poset $P$ is an acyclic category.