Iterated delooping and desuspension of structured ring 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 ${\tilde{\Sigma }}^r{\tilde{\Omega }}^r$-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 ${\tilde{\Omega }}^r{\tilde{\Sigma }}^r$-algebras.

17 pages

Cited by in corpus (3)

• On the Goodwillie derivatives of the identity in structured ring spectra
• Fibration theorems for TQ-completion of structured ring spectra
• TQ-completion and the Taylor tower of the identity functor