Ktheory of endomorphisms, Witt vectors, and cyclotomic spectra
Seminar/Forum
Of one the more interesting endofunctors of the category of categories is the one which associates to a category \(C\) its category of endomorphisms \(End(C)\). If \(C\) is a stable infinity category then \(End(C)\) is as well, and the associated Ktheory spectrum \(KEnd(C):=K(End(C))\) is called the Ktheory of endomorphisms of \(C\). Using calculations of Almkvist together with the theory of noncommutative motives of BlumbergGepnerTabuada, we classify equivalence classes of endomorphisms of the \(KEnd\) functor in terms of a noncompeleted version of the Witt vectors of the polynomial ring \(\mathbf{Z}[t]\), answering a question posed by Almkvist in the 70s. As applications, we obtain various lifts of Witt rings to the sphere spectrum as well as a more structured version of the cyclotomic trace via cyclic Ktheory, as studied in recent work of Kaledin and NikolausScholze.
Presenter

Dr David Gepner, Purdue University