## The universal property of topological Hochschild homology

Appears in collection : Chromatic Homotopy, K-Theory and Functors / Homotopie chromatique, K-théorie et foncteurs

Topological Hochschild homology is a fundamental invariant of rings and ring spectra, related to algebraic $K$-theory via the celebrated Dennis-Bökstedt trace map $K \rightarrow T H H$. Blumberg, Gepner and Tabuada showed that algebraic $K$-theory becomes especially well-behaved when considered as an invariant of stable $\infty$-categories, rather than just ring spectra: in that setting it can be described as the free additive invariant generated by the unit, that is, the initial additive functor under the the core $\infty$-groupoid functor, corepresented by the unit of $\mathrm{Cat}^{\mathrm{ex}}$. In this talk I will describe joint work with Thomas Nikolaus and Victor Saunier showing that $T H H$ similarly acquires a universal property when extended to stable $\infty$-categories, when one allows in addition to take coefficients in an arbitrary bimodule. In particular, we view $\mathrm{THH}$ as a functor on the category TCat ${ }^{\mathrm{ex}}$ whose objects are pairs $(C, M)$ where $C$ is a stable $\infty$-category and $M$ is a bimodule, that is, a biexact functor $C^{\mathrm{op}} \times C \rightarrow$ Spectra. We define a notion of being a trace-like invariant on TCat ${ }^{\mathrm{ex}}$, which amounts to sending certain maps in TCat ${ }^{\mathrm{ex}}$ to equivalences. We then show that $\mathrm{THH}$ is the free exact trace-like invariant generated from the unit of $\mathrm{TCat}^{\mathrm{ex}}$, where exact means exact in the bimodule entry. At the same time, algebraic $K$-theory can also be extended to to $\mathrm{TCat}^{\mathrm{ex}}$, in the form of endomorphism $K$-theory. Comparing universal properties we then get that $T H H$ is universally obtained from endomorphism $K$-theory by forcing exactness. This yields a conceptual proof that $T H H$ is the first Goodwillie derivative of endomorphism $K$-theory, and can be used to extend the Dundas-Goodwillie-McCarthy theorem to the setting of stable $\infty$-categories.