Morse theory for loop-free categories

Abstract

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].

There is an error. Moreover, the way to fix the error leads to the the better approach in the paper (which we did not know when we developed ours) Giacomo d’Antonio and Emanuele Delucchi, Minimality of toric arrangements, Journal of the European Mathematical Society (JEMS) 17 (2015), no. 3, 483–521. DOI: 10.4171/JEMS/508