Systems of Diagram Categories and K-theory. II

Systems of Diagram Categories and K-theory I

To any left system of diagram categories or to any left pointed derivateur (in the sense of Grothendieck) a K-theory space is associated. This K-theory space is shown to be canonically an infinite loop space and to have a lot of common properties with Waldhausen's K-theory. A weaker version of additivity is shown. Also, Quillen's K-theory of a large class of exact categories including the abelian categories is proved to be a retract of the K-theory of the associated derivateur.

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。

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…

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.

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.

连续K-理论的一些形式粘合图

Some formal gluing diagrams for continuous K-theory

本文构造刚性基上纤维-余纤维序列对应的对偶化稳定无穷范畴图，证明其在连续K-理论等局部化不变量下成为极限图，并借此恢复Clausen–Scholze的连续K-理论粘合及验证连续阿代尔下降。