On formality of diagrams of Eilenberg-MacLane spaces

TL;DR

This paper proves that diagrams of Eilenberg-MacLane spaces are formal over the rationals, leading to spectral sequence collapse, but not over other rings, using functor calculus.

Contribution

It establishes formality over $Q$ for diagrams of Eilenberg-MacLane spaces and shows non-formality over rings not containing $Q$.

Findings

Spectral sequence over $Q$ collapses at page 2 for diagrams of EML spaces.

Formality holds over $Q$ but not over rings lacking $Q$.

Abstract

In this paper, we establish formality (over $\mathbb{Q}$) for diagrams of Eilenberg-MacLane spaces of any height $n\geq 1$. This implies spectral sequence (over $\mathbb{Q}$) collapse at page $2$ for any diagram of EML spaces over any small category. We prove by functor calculus argument that formality does not hold over any fixed commutative ring $\mathbf{k}$ not containing $\mathbb{Q}$, where the category of diagrams is over the category generated by finite direct sums of a cyclic group.