geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
For ∞-actions of finite groups $G$ on objects $E$ in stable (∞,1)-categories, then the homotopy cofiber $X^{t G}$ of the norm map is called the Tate construction, sitting in a homotopy fiber sequence
(e.g Lurie, def. 6.1.6.24)
Depending on which edition you have, chapter 6 may be chapter 7.
For the stable (∞,1)-category of spectra this is accordingly called the Tate spectrum.
Tate spectra for cyclic groups play a role in the characterization of cyclotomic spectra.
For $N$ a $G$-representation and $H N$ its Eilenberg-MacLane spectrum, then the Tate spectrum $H N^{t G}$ represents what is classically called Tate cohomology
(Nikolaus-Scholze 17, p. 13) This may be generalized to possibly infinite discrete groups, see at Farrell-Tate cohomology.
The general abstract discussion is due to
Review with an eye towards topological cyclic homology is in
Last revised on June 21, 2022 at 16:38:24. See the history of this page for a list of all contributions to it.