[Paper Review] Iterated delooping and desuspension of structured ring spectra

Jacobson R. Blomquist|arXiv (Cornell University)|Oct 28, 2019

Homotopy and Cohomology in Algebraic Topology13 references4 citations

TL;DR

This paper establishes derived equivalences between structured ring spectra and their iterated suspension/desuspension constructions via homotopical comonads and monads. It proves that completion with respect to the iterated suspension functor yields a derived equivalence for 0-connected $Ο$-algebras and $r$-connected coalgebras over the associated comonad, and dually, cocompletion via iterated loops gives a derived equivalence for $r$-connected $Ο$-algebras and 0-connected algebras over the loop-suspension monad, using refined homotopical algebra in symmetric spectra.

ABSTRACT

We study completion with respect to the iterated suspension functor on $\mathcal{O}$-algebras, where $\mathcal{O}$ is a reduced operad in symmetric spectra. This completion is the unit of a derived adjunction comparing $\mathcal{O}$-algebras with coalgebras over the associated iterated suspension-loop homotopical comonad via the iterated suspension functor. We prove that this derived adjunction becomes a derived equivalence when restricted to 0-connected $\mathcal{O}$-algebras and $r$-connected $ ildeΣ^r ildeΩ^r$-coalgebras. We also consider the dual picture, using iterated loops to build a cocompletion map from algebras over the iterated loop-suspension homotopical monad to $\mathcal{O}$-algebras. This is the counit of a derived adjunction, which we prove is a derived equivalence when restricting to $r$-connected $\mathcal{O}$-algebras and $0$-connected $ ildeΩ^r ildeΣ^r$-algebras.

Motivation & Objective

• To understand how iterated suspension and desuspension operations on structured ring spectra relate to homotopy types via completion and cocompletion.
• To establish derived equivalences between $Ο$-algebras and coalgebras (or algebras) over iterated suspension-loop functors in the homotopical setting.
• To extend classical results like the Freudenthal suspension theorem and Bousfield-Kan completion to the context of $Ο$-algebras in symmetric spectra.
• To develop a proper homotopical framework using fibrant/cofibrant replacements and bar constructions to ensure derived adjunctions are well-behaved.

Proposed method

• Uses the iterated suspension functor $σ^r = σ^r C$ and loop functor $τ^r = τ^r F$ with fibrant and cofibrant replacements to ensure homotopical meaning.
• Constructs a derived adjunction between $Ο$-algebras and coalgebras over the iterated suspension-loop comonad via completion.
• Applies a cobar-type construction to model the resolution of algebras, ensuring proper (co)bar structures in the homotopical setting.
• Employs the positive flat stable model structure on $Ο$-algebras to define a homotopy theory and constructs a topological $A_∞$-category from mapping spaces.
• Uses higher Blakers-Massey theorems and connectivity estimates to prove cartesianness of cubes in the resolution, ensuring convergence of the completion process.
• Defines the homotopy category of $Ο$-algebras via path components of mapping spaces, and proves that a map is an isomorphism iff its underlying map is a weak equivalence.

Experimental results

Research questions

• RQ1 Under what conditions does completion with respect to the iterated suspension functor induce a derived equivalence between $Ο$-algebras and coalgebras over the associated comonad?
• RQ2 How does the dual process—cocompletion via iterated loops—relate to the homotopy theory of $Ο$-algebras?
• RQ3 What connectivity conditions ensure that the iterated suspension or loop maps are sufficiently highly connected to yield derived equivalences?
• RQ4 Can the classical Freudenthal suspension theorem be generalized to structured ring spectra in the context of $Ο$-algebras?
• RQ5 How do the homotopical bar constructions and (co)algebra structures behave in the derived category of $Ο$-algebras?

Key findings

• The derived adjunction between $Ο$-algebras and $σ^r τ^r$-coalgebras becomes a derived equivalence when restricted to 0-connected $Ο$-algebras and $r$-connected $σ^r τ^r$-coalgebras.
• The dual derived adjunction—cocompletion via the iterated loop-suspension monad—induces a derived equivalence between $r$-connected $Ο$-algebras and 0-connected $τ^r σ^r$-algebras.
• The connectivity of the dual Freudenthal map $F\tilde{\Sigma}^r \tilde{\Omega}^r X \to FX$ is $2k+3-r$ for $k$-connected $X$, providing a key estimate for convergence.
• The paper proves a higher Hurewicz theorem for structured ring spectra: if an $n$-cube is $({\rm id}+1)(k+1)$-cartesian, then its completion via $UQ$ is $({\rm id}+1)(k+1)$-cartesian.
• The completion process via $UQ$ preserves cartesianness under suitable connectivity assumptions, and the proof relies on higher Blakers-Massey theorems and cocartesianness estimates.
• A derived $Ο$-algebra map is an isomorphism in the homotopy category if and only if its underlying map $CA \to A'$ is a weak equivalence, linking algebraic structure to homotopy equivalence.