[Paper Review] Derived Koszul duality and TQ-homology completion of structured ring spectra

Michael Ching, John E. Harper|arXiv (Cornell University)|Feb 24, 2015

Homotopy and Cohomology in Algebraic Topology10 references6 citations

TL;DR

This paper establishes derived Koszul duality for structured ring spectra by showing that the adjunction between ${\mathcal{O}}$-algebra spectra and ${\mathsf{K}}$-coalgebra spectra via ${\mathsf{TQ}}$-homology becomes an equivalence when restricted to $0$-connected ${\mathcal{O}}$-algebras. The key result resolves the $0$-connected case of a conjecture by Francis-Gaitsgory and establishes a spectral analog of Quillen and Sullivan's rational homotopy theory.

ABSTRACT

Working in the context of symmetric spectra, we consider any higher algebraic structures that can be described as algebras over an operad O. We prove that the fundamental adjunction comparing O-algebra spectra with coalgebra spectra over the associated comonad K, via topological Quillen homology (or TQ-homology), can be turned into an equivalence of homotopy theories by replacing O-algebras with the full subcategory of 0-connected O-algebras. This resolves in the affirmative the 0-connected case of a conjecture of Francis-Gaitsgory. This derived Koszul duality result can be thought of as the spectral algebra analog of the fundamental work of Quillen and Sullivan on the rational homotopy theory of spaces, and the subsequent p-adic and integral work of Goerss and Mandell on cochains and homotopy type---the following are corollaries of our main result: (i) 0-connected O-algebra spectra are weakly equivalent if and only if their TQ-homology spectra are weakly equivalent as derived K-coalgebras, and (ii) if a K-coalgebra spectrum is 0-connected and cofibrant, then it comes from the TQ-homology spectrum of an O-algebra. We construct the spectral algebra analog of the unstable Adams spectral sequence that starts from the TQ-homology groups TQ_*(X) of an O-algebra X, and prove that it converges strongly to pi_*(X) when X is 0-connected.

Motivation & Objective

• To resolve the $0$-connected case of a conjecture by Francis-Gaitsgory on derived Koszul duality for structured ring spectra.
• To establish an equivalence of homotopy theories between $0$-connected ${\mathcal{O}}$-algebra spectra and their ${\mathsf{TQ}}$-homology spectra as ${\mathsf{K}}$-coalgebras.
• To construct a spectral analog of the unstable Adams spectral sequence that converges strongly to $\pi_*(X)$ for $0$-connected ${\mathcal{O}}$-algebras $X$.
• To provide a homotopical framework for understanding completion and homotopy type via ${\mathsf{TQ}}$-homology in the context of structured ring spectra.
• To extend foundational results from rational and $p$-adic homotopy theory—such as Quillen and Sullivan theory—into the setting of $E_n$-ring spectra and operadic algebras over spectra.

Proposed method

• Work within the category of symmetric spectra to define and study ${\mathcal{O}}$-algebra spectra for operads ${\mathcal{O}}$.
• Use topological Quillen homology (${\mathsf{TQ}}$-homology) as the derived abelianization functor, defined as the total left derived functor of the indecomposable quotient.
• Construct the comonad ${\mathsf{K}}$ on ${\mathcal{O}}$-algebras via the adjunction between ${\mathcal{O}}$-algebras and their ${\mathsf{TQ}}$-homology spectra.
• Prove that the adjunction between ${\mathcal{O}}$-algebras and ${\mathsf{K}}$-coalgebras becomes a Quillen equivalence when restricted to $0$-connected ${\mathcal{O}}$-algebras.
• Utilize totalization and homotopy limit techniques to analyze cosimplicial resolutions, particularly $\operatorname{Tot}$ and $\operatorname{holim}_{\Delta}$, to compare homotopy types.
• Establish strong convergence of the unstable Adams spectral sequence starting from ${\mathsf{TQ}}_*(X)$ to $\pi_*(X)$ for $0$-connected $X$ via Reedy fibrant replacement and Quillen equivalence arguments.

Experimental results

Research questions

• RQ1 Can the adjunction between ${\mathcal{O}}$-algebra spectra and ${\mathsf{K}}$-coalgebra spectra via ${\mathsf{TQ}}$-homology be turned into a Quillen equivalence?
• RQ2 Does the derived Koszul duality framework for structured ring spectra hold in the $0$-connected setting, as conjectured by Francis-Gaitsgory?
• RQ3 Can a spectral analog of the unstable Adams spectral sequence be constructed that converges strongly to $\pi_*(X)$ for $0$-connected ${\mathcal{O}}$-algebras $X$?
• RQ4 To what extent do ${\mathsf{TQ}}$-homology spectra classify $0$-connected ${\mathcal{O}}$-algebra spectra up to weak equivalence?
• RQ5 How do the homotopical properties of ${\mathsf{TQ}}$-homology and the associated comonad ${\mathsf{K}}$ reflect the algebraic-topological structure of $E_n$-ring spectra and operadic algebras?

Key findings

• The adjunction between ${\mathcal{O}}$-algebra spectra and ${\mathsf{K}}$-coalgebra spectra becomes a Quillen equivalence when restricted to the full subcategory of $0$-connected ${\mathcal{O}}$-algebras.
• A $0$-connected ${\mathcal{O}}$-algebra spectrum $X$ is weakly equivalent to another if and only if their ${\mathsf{TQ}}$-homology spectra are weakly equivalent as derived ${\mathsf{K}}$-coalgebras.
• Every $0$-connected and cofibrant ${\mathsf{K}}$-coalgebra spectrum arises as the ${\mathsf{TQ}}$-homology spectrum of some ${\mathcal{O}}$-algebra.
• The unstable Adams spectral sequence starting from ${\mathsf{TQ}}_*(X)$ converges strongly to $\pi_*(X)$ for any $0$-connected ${\mathcal{O}}$-algebra $X$.
• The construction of ${\mathsf{TQ}}$-homology as a derived indecomposables functor provides a spectral analog of Quillen’s rational homotopy theory and Goerss–Mandell’s $p$-adic theory.
• The proof relies on advanced homotopical techniques, including totalization, Reedy fibrant replacement, and Quillen equivalences between simplicial sets and compactly generated Hausdorff spaces, to establish the required weak equivalences.