K-theory of endomorphisms, Witt vectors, and cyclotomic spectra

David Gepner – 13 June 2018

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 K-theory spectrum KEnd(C):=K(End(C)) is called the K-theory of endomorphisms of C. Using calculations of Almkvist together with the theory of noncommutative motives of Blumberg-Gepner-Tabuada, we classify equivalence classes of endomorphisms of the KEnd functor in terms of a noncompeleted version of the Witt vectors of the polynomial ring 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 K-theory, as studied in recent work of Kaledin and Nikolaus-Scholze.