Iterated delooping and desuspension of structured ring spectra

2019/10/28 by Jacobson R. Blomquist, Blomquist, Jacobson R.
Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1910.12442

openalex publication_date 2019/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study completion with respect to the iterated suspension functor on O-algebras, where O is a reduced operad in symmetric spectra. This completion is the unit of a derived adjunction comparing 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 O-algebras and r-connected Σ^r Ω^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 O-algebras. This is the counit of a derived adjunction, which we prove is a derived equivalence when restricting to r-connected O-algebras and 0-connected Ω^r Σ^r-algebras.

Citations

• Genetics of Homotopy Theory and the Adams Conjecture

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