# Is there discrete Morse theory on acyclic categories?

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.

• You might be interested in this paper, which describes morse theory on a certain class of acyclic poset-enriched categories via localization: arxiv.org/abs/1510.01907 If this is the sort of thing you are looking for, let me know and I can describe it in an answer. Sep 3, 2020 at 22:57