| Popis: |
We study completion with respect to the iterated suspension functor on 풪-algebras, where 풪 is a reduced operad in symmetric spectra. It turns out it is the unit of a derived adjunction comparing 풪-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 풪-algebras and r connected Σr Ωr -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 풪-algebras. This is the counit of a derived adjunction, which we prove is a derived equivalence when restricting to r-connected 풪-algebras and 0-connected Ωr Σr-algebras. The final chapter also goes through similar analysis in the context of spaces, for iterated suspension, stabilization, and integral chains, as well as the dual picture for iterated loops. |
