Systems of Diagram Categories and K-theory I

Systems of Diagram Categories and K-theory. II

The additivity theorem for derivateurs associated to complicial biWaldhausen categories is proved. Also, to any exact category in the sense of Quillen a K-theory space is associated. This K-theory is shown to satisfy the additivity, approximation and resolution theorems.

丰富 Mackey 函子的同伦理论

Homotopy Theory of Enriched Mackey Functors

本文系统发展了闭多范畴上富集的图表、预层与 Mackey 函子理论，研究富集改变及其复合性，并将其应用于由 K-理论多函子诱导的同伦理论等价，涵盖带点多范畴、置换范畴和对称谱等情形。

在带点稳定双Segal空间中识别CGW范畴

Recognizing CGW categories among pointed stable double Segal spaces

研究CGW范畴与带点稳定双Segal空间的关系，表明CGW范畴可视为带点稳定双Segal空间，通过双范畴分类图构造来识别二者，为代数K理论提供了新视角。

The bivariant parasimplicial $\mathsf{S}_{\bullet}$-construction

Coherent strings of composable morphisms play an important role in various important constructions in abstract stable homotopy theory (for example algebraic K-theory or higher Toda brackets) and in the representation theory of finite dimensional algebras (as representations of Dynkin quivers of type A). In a first step we will prove a strong comparison result relating composable strings of morphisms and coherent diagrams on cubes with support on a path from the initial to the final object. We observe that both structures are equivalent (by passing to higher analogues of mesh categories) to dis…

Milnor descent for cohesive dg-categories

We show that the functor from curved differential graded algebras to differential graded categories, defined by the second author in [B], sends Cartesian diagrams to homotopy Cartesian diagrams, under certain reasonable hypotheses. This is an extension to the arena of dg categories of a construction of projective modules due to Milnor. As an example, we show that the functor satisfies descent for certain partitions of a complex manifold.

Quasi DG categories and mixed motivic sheaves

We introduce the notion of a quasi DG category, generalizing that of a DG category. To a quasi DG category satisfying certain additional conditions, we associate another quasi DG category, the quasi DG category of $C$-diagrams. We then show the homotopy category of the quasi DG category of $C$-diagrams has the structure of a triangulated category. This procedure is then applied to produce a triangulated category of mixed motives over a base variety.