The homotopy theory of cyclotomic spectra

Abstract

We describe spectral model category structures on the categories of cyclotomic spectra and $p$–cyclotomic spectra (in orthogonal spectra) with triangulated homotopy categories. We show that the functors $\mathit{TR}$ and $\mathit{TC}$ are corepresentable in these categories. Specifically, the derived mapping spectrum out of the sphere spectrum in the category of cyclotomic spectra corepresents the finite completion of $\mathit{TC}$ and the derived mapping spectrum out of the sphere spectrum in the category of $p$–cyclotomic spectra corepresents the $p$–completion of $\mathit{TC}\left(-;p\right)$.