Koszulity of cohomology = $K(\pi,1)$-ness + quasi-formality

TL;DR

This paper explores the relationship between Koszulity, $K(\pi,1)$-spaces, and quasi-formality in cohomology, providing theoretical insights, categorical interpretations, and counterexamples to previous conjectures.

Contribution

It introduces a categorical perspective on Koszulity and $K(\pi,1)$-spaces, discusses Massey operations, and applies Koszul duality to establish main results, including a counterexample to formality.

Findings

Koszulity implies $K(\pi,1)$-ness under certain conditions

Quasi-formality is characterized by vanishing Massey multiplications

Counterexample shows not all cochain DG-algebras are formal

Abstract

This paper is a greatly expanded version of Section 9.11 in arXiv:1006.4343. A series of definitions and results illustrating the thesis in the title (where quasi-formality means vanishing of a certain kind of Massey multiplications in the cohomology) is presented. In particular, we include a categorical interpretation of the "Koszulity implies $K(\pi,1)$" claim, discuss the differences between two versions of Massey operations, and apply the derived nonhomogeneous Koszul duality theory in order to deduce the main theorem. In the end we demonstrate a counterexample providing a negative answer to a question of Hopkins and Wickelgren about formality of the cochain DG-algebras of absolute Galois groups, thus showing that quasi-formality cannot be strengthened to formality in the title assertion.