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 Ώ(BG∧p), whenGis a finite group,BG∧p is the p-completion of its classifying space, and Ώ(BG∧p)is the loop space ofBG∧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 C is a small category, |C| is the geometric realization of its nerve, R is a commutative ring, and |C|+R is a “plus construction” for |C| in the sense of Quillen (taken with respect to R-homology), then H∗(Ω(|C|+R); R) can be described as the homology of a chain complex of projective RC-modules satisfying a certain list of algebraic conditions that determine it uniquely up to chain homotopy. Benson’s theorem is now the case where C is the category of a finite group G, R = Fp for some prime p, and |C|+R=BG∧p.. © 2021, Mathematical Science Publishers. All rights reserved.