Loop space homology of a small category

Abstract

In a 2009 paper, Dave Benson gave a description in purely algebraic terms of the mod $p$ homology of $\Omega(BG^\wedge_p)$, when $G$ is a finite group, $BG^\wedge_p$ is the $p$-completion of its classifying space, and $\Omega(BG^\wedge_p)$ is the loop space of $BG^\wedge_p$. The main purpose of this work is to shed new light on Benson's result by extending it to a more general setting. As a special case, we show that if $\mathcal{C}$ is a small category, $|\mathcal{C}|$ is the geometric realization of its nerve, $R$ is a commutative ring, and $|\mathcal{C}|^+_R$ is a "plus construction" for $|\mathcal{C}|$ in the sense of Quillen (taken with respect to $R$-homology), then $H_*(\Omega(|\mathcal{C}|^+_R);R)$ can be described as the homology of a chain complex of projective $R\mathcal{C}$-modules satisfying a certain list of algebraic conditions that determine it uniquely up to chain homotopy. Benson's theorem is now the case where $\mathcal{C}$ is the category of a finite group $G$, $R=\mathbb{F}_p$ for some prime $p$, and $|\mathcal{C}|^+_R=BG^\wedge_p$.