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

## Seminar/Forum

107
Peter Hall
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 $$\mathbf{Z}[t]$$, 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 K-theory, as studied in recent work of Kaledin and Nikolaus-Scholze.