#### Volume 19, issue 6 (2015)

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The homotopy theory of cyclotomic spectra

### Andrew J Blumberg and Michael A Mandell

Geometry & Topology 19 (2015) 3105–3147
##### 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 $\mathit{TC}\left(-;p\right)$.

##### Keywords
topological cyclic homology, cyclotomic spectrum, model category, ABC category
##### Mathematical Subject Classification 2010
Primary: 19D55
Secondary: 18G55, 55Q91
##### Publication
Received: 23 May 2013
Revised: 2 March 2015
Accepted: 26 April 2015
Published: 6 January 2016
Proposed: Mark Behrens
Seconded: Bill Dwyer, Haynes Miller
##### Authors
 Andrew J Blumberg Department of Mathematics The University of Texas Austin, TX 78712 USA Michael A Mandell Department of Mathematics Indiana University Rawles Hall 831 E 3rd St Bloomington, IN 47405 USA