Generalized spectral categories, topological Hochschild homology and trace maps

Abstract

Given a monoidal model category C and an object K in C , Hovey constructed in [22] the monoidal model category Sp † .C; K/ of K -symmetric spectra over C . In this paper we describe how to lift a model structure on the category of C -enriched categories to the category of Sp † .C; K/-enriched categories. This allow us to construct a (four step) zig-zag of Quillen equivalences comparing dg categories to H Z-categories.As an application we obtain: (1) the invariance under weak equivalences of the topological Hochschild homology (THH ) and topological cyclic homology (TC ) of dg categories; (2) non-trivial natural transformations from algebraic K -theory to THH . 55P42, 18D20, 18G55; 19D55

Dwyer-Kan homotopy theory of enriched categories

“…They were later used by Shipley [26] to establish a zig-zag of Quillen equivalences between differential graded k-algebras and algebras over the Eilenberg-MacLane ring spectrum Hk. Hence, the following results of Tabuada [30,31] can be recovered from Theorem 1.4 in a unified way. Corollary 1.6.…”

An example of a non-Fourier–Mukai functor between derived categories of coherent sheaves

“…In [51] (following [49]) Tabuada constructs an equivalence 6 H : Ho(dgcat(R)) → Ho(Cat Sp (HR)). The construction of H is quite involved but it is not hard to verify that for a R-linear DG-category c one has an R-linear equivalence between π 0 H(c) and H 0 (c).…”

Fundamental isomorphism conjecture via non‐commutative motives

“…A first observation is that all the functors \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbf {E}}:{\mathrm{Or}}(G)\rightarrow {\mathcal {M}}$\end{document} which appear in our original examples have in common that they actually factor, not only via R ‐linear categories, but via their associated dg‐category : For non‐connective K ‐theory, one can use Schlichting's construction 16, Section 6.4]; for Hochschild and cyclic homology, see 8, Section 5.3]; for topological Hochschild homology, see 3, Section 3] or 20, Section 8.1]. Let us quickly recall basic facts about this category \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathsf {dgcat}}$\end{document}.…”

“…This means that the functor \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$E:{\mathsf {dgcat}} \rightarrow {\mathcal {M}}$\end{document} preserves filtered homotopy colimits and final object and that E is such that every (Drinfeld) short exact sequence of dg categories is mapped to a distinguished triangle in the homotopy category \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathsf {Ho}}({\mathcal {M}})$\end{document}. Thanks to the work of Waldhausen 22 and Schlichting 16, of Weibel 23, of Keller 7 and of Blumberg‐Mandell 3 (see also 20), all the above classical theories are localizing invariants. It will be convenient to have a short name for the induced functors on the orbit category.…”