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The universal property of topological Hochschild homology

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Auteurs : Harpaz, Yonatan (Auteur de la Conférence)
CIRM (Editeur )

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Résumé : 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.

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    Informations sur la Vidéo

    Réalisateur : Petit, Jean
    Langue : Anglais
    Date de publication : 17/02/2023
    Date de captation : 24/01/2023
    Sous collection : Research talks
    arXiv category : Algebraic Topology ; Analysis of PDEs
    Domaine : Topology
    Format : MP4 (.mp4) - HD
    Durée : 01:05:50
    Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students
    Download : https://videos.cirm-math.fr/2023-01-24-harpaz.mp4

Informations sur la Rencontre

Nom de la rencontre : Chromatic Homotopy, K-Theory and Functors / Homotopie chromatique, K-théorie et foncteurs
Organisateurs de la rencontre : Ausoni, Christian ; Hess Bellwald, Kathryn ; Powell, Geoffrey ; Touzé, Antoine ; Vespa, Christine
Dates : 23/01/2023 - 27/01/2023
Année de la rencontre : 2023
URL Congrès : https://conferences.cirm-math.fr/2339.html

Données de citation

DOI : 10.24350/CIRM.V.19997303
Citer cette vidéo: Harpaz, Yonatan (2023). The universal property of topological Hochschild homology. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19997303
URI : http://dx.doi.org/10.24350/CIRM.V.19997303

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