Higher traces, noncommutative motives, and the categorified Chern character
type: Item
instance of: scholarly article
title: Higher traces, noncommutative motives, and the categorified Chern character
zbMATH Open document ID: 1361.14014
author: Marc Hoyois
author: Sarah Scherotzke
author: Nicolò Sibilla
published in: Advances in Mathematics
publication date: 2017-02-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1511.03589
full work available at URL: final-v3.pdf
review text: The authors propose a categorification of the Chern character as a certain symmetric monoidal functor from a category of mixed non commutative motives on an algebraic stack \(X\) to the category of \(S^1\)-equivariant perfect complexes on the loop stack of \(X\). These results follow from some general statement about higher categorical traces in symmetric monoidal \((\infty,n)\)-categories. First, the authors develop a general formalism for higher traces in symmetric monoidal \((\infty,n)\)-categories showing that they satisfy similar statements known for the topological Hochschild homology of a small stable \((\infty,1)\)-category with coefficients in a bimodule over it. Second, they provide an interpretation of the connective and non-connective \(K\)-theory spectra of a small, stable, and idempotent complete \((\infty,1)\)-category \(\mathcal{A}\) tensored over a rigid, stable and small symmetric monoidal \((\infty,1)\)-category \(\mathcal{E}\) in terms of a certain spectra of maps in a stable higher category of motives. More precisely, they construct stable and presentable \((\infty,1)\)-categories \(Mot(\mathcal{E})\) and \(\mathbb{M}ot(\mathcal{E})\) of additive and localizing \(\mathcal{E}\)-motives, they show that these are the target of two universal functors \(\mathcal{U}_{\mathrm{add}}\) and \(\mathcal{U}_{\mathrm{loc}}\) from the \((\infty,1)\)-category \(Cat^{\mathrm{perf}}(\mathcal{E})\) of small, stable, and idempotent complete \((\infty,1)\)-categories tensored over \(\mathcal{E}\) and that the \(K\)-theoretic spectra of \(\mathcal{A}\) before mentioned are identified with the spectra of morphisms in \(Mot(\mathcal{E})\) (respectively in \(\mathbb{M}ot(\mathcal{E})\)) from \(\mathcal{U}_{\mathrm{add}}(\mathcal{E})\) into \(\mathcal{U}_{\mathrm{add}}(\mathcal{A})\) (respectively from \(\mathcal{U}_{\mathrm{loc}}(\mathcal{E})\) into \(\mathcal{U}_{\mathrm{loc}}(\mathcal{A})\)). This is an adaptation of the results of Blumberg, Gepner and Tabuada on high…
reviewed by: Giovanni Faonte
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 18D05
Mathematics Subject Classification ID: 19D55
zbMATH DE Number: 6686567
zbMATH Keywords: traces
zbMATH Keywords: noncommutative motives
zbMATH Keywords: Chern characters
zbMATH Keywords: secondary \(K\)-theory
MaRDI profile type: Publication
OpenAlex ID: W2964290142
arXiv ID: 1511.03589
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DOI: 10.1016/J.AIM.2017.01.008
description: scientific article