  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? [14].
