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 $|\\mathcal{C}|^+_R=BG^\\wedge_p$.