# Loop space homology of a small category

Carles Broto, Ran Levi, Bob Oliver

Research output: Working paper

### 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 $|\mathcal{C}|^+_R=BG^\wedge_p$.
Original language English ArXiv Submitted - 6 Jul 2018

### Fingerprint

Loop Space
Wedge
Homology
Finite Group
Classifying Space
Nerve
Commutative Ring
Homotopy
Completion
Module
Term
Theorem

• math.AT
• 55R35

### Cite this

Broto, C., Levi, R., & Oliver, B. (2018). Loop space homology of a small category. ArXiv.

Loop space homology of a small category. / Broto, Carles; Levi, Ran; Oliver, Bob.

ArXiv, 2018.

Research output: Working paper

Broto, C, Levi, R & Oliver, B 2018 'Loop space homology of a small category' ArXiv.
Broto C, Levi R, Oliver B. Loop space homology of a small category. ArXiv. 2018 Jul 6.
Broto, Carles ; Levi, Ran ; Oliver, Bob. / Loop space homology of a small category. ArXiv, 2018.
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