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 $TC(-;p)$.