Operations on spectral partition Lie algebras and TAQ cohomology

TL;DR

The paper classifies all natural operations on the homotopy groups of spectral partition Lie algebras over $\mathbb{F}_p$ and on reduced mod $p$ TAQ cohomology, showing these operations are governed by a universal set of weight $p^k$ unary operations and a shifted Lie bracket with a nontrivial restriction. Using a dual bar spectral sequence and algebraic Koszul duality, the authors identify the weight $p^k$ operations with unstable Ext groups over the Dyer–Lashof algebra and organize them into a power ring $\mathcal{P}$ whose Yoneda-type product encodes composition. They construct a shifted restricted Lie algebra structure that interacts with the unary operations via explicit Adem and Nishida type relations, and they prove that these structures generate all natural operations, recovering and clarifying unpublished results on TAQ cohomology. A homotopy fixed points spectral sequence detects the restriction and shows compatibility with the shifted Lie bracket, leading to a complete algebraic framework for the mod $p$ TAQ cohomology of $\mathbb{E}_\infty$-algebras. The results establish a robust Koszul dual description of the operation theory and provide explicit bases and relations that match the known ranks of free objects, with corollaries for $\mathbb{S}$-linear TAQ cohomology operations.

Abstract

We determine all natural operations and their relations on the homotopy groups of spectral partition Lie algebras, which coincide with $\mathbb{F}_p$-linear topological André-Quillen cohomology operations at any prime. We construct unary operations and a shifted restricted Lie algebra structure on the homotopy groups of spectral partition Lie algebras. Then we prove a composition law for the unary operations, as well as a compatibility condition between unary operations and the shifted Lie bracket with restriction up to a unit for the restriction. Comparing with Brantner-Mathew's result on the ranks of the homotopy groups of free spectral partition Lie algebras, we deduce that these generate all natural operations, thereby also recovering unpublished results of Kriz and Basterra-Mandell on $\mathbb{F}_p$-linear TAQ cohomology operations. As a corollary, we determine the structure of natural operations on mod $p$ $\mathbb{S}$-linear TAQ cohomology.

Theorems & Definitions (81)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Theorem 1.6
• Remark 1.7
• Definition 2.1
• Proposition 2.2
• Definition 2.3