Multiplicative Structures on Algebraic K-Theory

The algebraic $K$-theory of Waldhausen $\infty$-categories is the functor corepresented by the unit object for a natural symmetric monoidal structure. We therefore regard it as the stable homotopy theory of homotopy theories. In particular, it respects all algebraic structures, and as a result, we obtain the Deligne Conjecture for this form of $K$-theory

Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin

With an explicit, algebraic indexing $(2,1)$-category, we develop an efficient homotopy theory of cyclonic objects: circle-equivariant objects relative to the family of finite subgroups. We construct an $\infty$-category of cyclotomic spectra as the homotopy fixed points of an action of the multiplicative monoid of the natural numbers on the category of cyclonic spectra. Finally, we elucidate and prove a conjecture of Kaledin on cyclotomic complexes.Comment: 28 pages. Comments very welcom

Regularity of structured ring spectra and localization in K-theory

We identify a regularity property for structured ring spectra, and with it we prove a natural analogue of Quillen's localization theorem for algebraic K-theory in this setting.Comment: 9 pages. Corrected references and aspects of the expositio