The homotopy theory of cyclotomic spectra

TL;DR

This paper develops a homotopy-theoretic framework for cyclotomic spectra, establishing model structures and corepresentability of key functors, advancing the understanding of their algebraic and topological properties.

Contribution

It introduces spectral model category structures on cyclotomic spectra and shows that the functors TR and TC are corepresentable within these categories.

Findings

Spectral model structures on cyclotomic spectra established.

TR and TC functors are corepresentable in these categories.

Derived mapping spectra relate to completions of TC.

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 $TR$ 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 $TC(-;p)$.