Higher traces, noncommutative motives, and the categorified Chern character
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 higher algebraic \(K\)-theory [\textit{A. J. Blumberg} et al., Geom. Topol. 17, No. 2, 733--838 (2013; Zbl 1267.19001)] to \(\mathcal{E}\)-linear \((\infty,1)\)-categories. They then apply these general results to extend the well known construction of the Chern character. In fact, the Chern character can be seen as a morphisms from the \(K\)-theory of a derived stack \(X\) into \(H^{\mathrm{per}}_{dR}(X)\) the \(2\)-periodization of the De Rham complex of \(X\) over the base filed \(k\). The latter can be identified with the \(E_{\infty}\)-ring spectra of \(S^1\)-equivariant functions on the loop stack of \(X\) localized at the generator of the cohomology of the classifying stack \(B S^1\). \textit{B. Toën} and \textit{G. Vezzosi} [Sel. Math., New Ser. 21, No. 2, 449--554 (2015; Zbl 1333.19006)] developed a formalism for secondary Chern characters and \(K\)-theory and related these constructions with the double loop stack of \(X\). In this sense, their work aims to show that the classical Chern character and \(K\)-theory are only the first layer of a much richer and refined class of invariants. In their framework the secondary Chern character is a morphism from \(i_0(Cat^{sat}(X))\), the \(\infty\)-groupoid of dualizable categorical sheaves on X, to \(\Omega^{\infty}\mathcal{O}(\mathcal{L}^{2} X)^{h(S^1\times S^1)}\). The main application of the more general results of this paper is an enhancement of the statement of Toën and Vezzosi. The authors show that the secondary Chern character of Toën and Vezzosi can be promoted to a symmetric monoidal \((\infty,1)\)-functor from \(i_1(Cat(Perf(X)))\), the maximal sub-\((\infty,1)\)-category of the \((\infty,2)\)-category of small stable \((\infty,1)\)-categories linear over \(Perf(X)\) (the category of perfect complexes over \(X\)), to the \((\infty,1)\)-category \(QCoh(\mathcal{L}(X))\) of quasi-coherent complexes on the loop stack of \(X\). Moreover, this functor descend to a symmetric monoidal functor from \(\mathbb{M}ot(Perf(X))\) to the \(S^1\)-equivariant part of \(QCoh(\mathcal{L}(X))\). Finally, the authors define their version of secondary \(K\)-theory \(\mathbb{K}^{(2)}(X)\) which is canonically mapped into by the secondary \(K\)-theory of Toën and Vezzosi, and use this last mentioned result to show that the secondary Chern character is refined by a morphism of \(E_{\infty}\)-ring spectra from \(\mathbb{K}^{(2)}(X)\) to \(\mathcal{O}(\mathcal{L}^{2} X)^{h(S^1\times S^1)}\). Despite the highly technical and intense profile of this work, which is essentially addressed to very specialists in the field, the results provided do show originality and provide an improvement and possibly a more general framework for the higher analogues of Chern characters and \(K\)-theory.
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