TITLE: Is there discrete Morse theory on acyclic categories?
QUESTION [7 upvotes]: 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.

REPLY [5 votes]: Recently, it appeared in ArXiv (https://arxiv.org/abs/2107.06202) a paper which may be related to this question. The title is:

Morse theory for loop-free categories

Note that loop-free categories is the same as acyclic categories. The abstract is:

We extend discrete Morse-Bott theory to the setting of loop-free (or
acyclic) categories. First of all, we state a homological version of
Quillen's Theorem A in this context and introduce the notion of
cellular categories. Second, we present a notion of vector field for
loop-free categories. Third, we prove a homological collapsing theorem
in the absence of critical objects in order to obtain the Morse
inequalities. Examples are provided through the exposition. This
answers partially a question by T. John: whether there is a Morse
theory for loop-free (or acyclic) categories?