The homotopy theory of cyclotomic spectra

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 TR and 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 TC and the derived mapping spectrum out of the sphere spectrum in the category of p -cyclotomic spectra corepresents the p -completion of TC.I p/.