  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 $TC(-;p)$.
