1.5. Related works
The “motivic” idea of constructing universal invariants is not new and appears in several different subjects: for example, Cortiñas-Thom’s work [22] on bivariant algebraic -theory, Higson’s work [41] on Kasparov’s bivariant -theory, Meyer-Nest’s work [60] on -algebras, Morel-Voevodsky’s work [59] on -homotopy theory of schemes, and Voevodsky’s work [82] on (mixed) motives.
In this vein, over the past few years the third author and Cisinski have carried out a program [20, 21, 73] of providing universal characterizations of algebraic -theory in the setting of dg-categories, using the formalism of derivators. One of the main goals of our work in this paper completes this program by extending these results to stable -categories and by solving a key open question left open in the previous work of the third author, namely identifying the cyclotomic trace in terms of the co-representability results.
Finally, we would like to mention that Barwick [3] has recent work on a universal characterization of higher algebraic -theory, in the context of a detailed study of the algebraic -theory of -categories.
Acknowledgments: The authors would like to thank Mike Mandell for many helpful conversations and Haynes Miller for asking motivating questions. They are grateful to David Ben-Zvi, Chris Brav, Bob Bruner, Jonathan Campbell, Denis-Charles Cisinski, Bjorn Dundas, Tom Goodwillie, Jeremiah Heller, Kathryn Hess, John Lind, Peter May, Jack Morava, Markus Spitzweck and Bertrand Toën for helpful comments on a previous draft. The authors would like also to thank the anonymous referees for very careful readings and detailed comments and corrections which substantially improved this paper. This project was initiated during a visit by the second and third authors to Stanford’s math department, and they would like to thank the department for its hospitality. Finally, the authors would like to thank the Midwest Topology Network for funding various trips which facilitated the conduct of this research. A. J. Blumberg was supported in part by NSF grant DMS-0906105. G. Tabuada was supported by the Estimulo à Investigação Award 2008 - Calouste Gulbenkian Foundation, by the FCT-Portugal grants PTDC/MAT/098317/2008 and SFRH/BSAB/1116/2011 and by the NEC award-2742738.
Original source: arXiv:1001.2282v4