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Register a new userSpectral Waldhausen categories, the $S_bullet$-construction, and the Dennis trace
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We give an explicit point-set construction of the Dennis trace map from the $K$-theory of endomorphisms $Kmathrm{End}(mathcal{C})$ to topological Hochschild homology $mathrm{THH}(mathcal{C})$ for any spectral Waldhausen category $mathcal{C}$. We describe the necessary technical foundations, most notably a well-behaved model for the spectral category of diagrams in $mathcal{C}$ indexed by an ordinary category via the Moore end. This is applied to define a version of Waldhausens $S_{bullet}$-construction for spectral Waldhausen categories, which is central to this account of the Dennis trace map. Our goals are both convenience and transparency---we provide all details except for a proof of the additivity theorem for $mathrm{THH}$, which is taken for granted---and the exposition is concerned not with originality of ideas, but rather aims to provide a useful resource for learning about the Dennis trace and its underlying machinery.
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It is known by results of Dyckerhoff-Kapranov and of Galvez--Carrillo-Kock-Tonks that the output of the Waldhausen S.-construction has a unital 2-Segal structure. Here, we prove that a certain S.-functor defines an equivalence between the category of augmented stable double categories and the category of unital 2-Segal sets. The inverse equivalence is described explicitly by a path construction. We illustrate the equivalence for the known examples of partial monoids, cobordism categories with genus constraints and graph coalgebras.
In a previous paper, we showed that a discrete version of the $S_bullet$-construction gives an equivalence of categories between unital 2-Segal sets and augmented stable double categories. Here, we generalize this result to the homotopical setting, by showing that there is a Quillen equivalence between a model category for unital 2-Segal objects and a model category for augmented stable double Segal objects which is given by an $S_bullet$-construction. We show that this equivalence fits together with the result in the discrete case and briefly discuss how it encompasses other known $S_bullet$-constructions.
In previous work, we develop a generalized Waldhausen $S_{bullet}$-construction whose input is an augmented stable double Segal space and whose output is a unital 2-Segal space. Here, we prove that this construction recovers the previously known $S_{bullet}$-constructions for exact categories and for stable and exact $(infty,1)$-categories, as well as the relative $S_{bullet}$-construction for exact functors.
We show that the characteristic polynomial and the Lefschetz zeta function are manifestations of the trace map from the $K$-theory of endomorphisms to topological restriction homology (TR). Along the way we generalize Lindenstrauss and McCarthys map from $K$-theory of endomorphisms to topological restriction homology, defining it for any Waldhausen category with a compatible enrichment in orthogonal spectra. In particular, this extends their construction from rings to ring spectra. We also give a revisionist treatment of the original Dennis trace map from $K$-theory to topological Hochschild homology (THH) and explain its connection to traces in bicategories with shadow (also known as trace theories).
We prove the equivalence of several hypotheses that have appeared recently in the literature for studying left Bousfield localization and algebras over a monad. We find conditions so that there is a model structure for local algebras, so that localization preserves algebras, and so that localization lifts to the level of algebras. We include examples coming from the theory of colored operads, and applications to spaces, spectra, and chain complexes.
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