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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.

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    $\begingroup$ 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. $\endgroup$ Sep 3, 2020 at 22:57

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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?

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