A note on cohomological localization

Localizations of the category of $A_\infty$ categories and internal Homs

We prove that the localizations of the categories of dg categories, of cohomologically unital and strictly unital $A_\infty$ categories with respect to the corresponding classes of quasi-equivalences are all equivalent. Moreover we show that the last two localizations are equivalent to the corresponding quotients by the relation of being isomorphic in the cohomology of the $A_\infty$ category of $A_\infty$ functors. As an application we give a complete proof of a claim by Kontsevich stating that the category of internal Homs for two dg categories can be described as the category of strictly un…

Localization theory for triangulated categories

These notes provide an introduction to the theory of localization for triangulated categories. Localization is a machinery to formally invert morphisms in a category. We explain this formalism in some detail and we show how it is applied to triangulated categories.

A note on K-theory and triangulated derivators

In this paper we show an example of two differential graded algebras that have the same derivator K-theory but non-isomorphic Waldhausen K-theory. We also prove that Maltsiniotis's comparison and localization conjectures for derivator K-theory cannot be simultaneously true.

三角范畴相对于扩张闭子范畴的局部化

Localization of triangulated categories with respect to extension-closed subcategories

本文构建三角范畴相对于扩张闭子范畴的外三角局部化框架，统一了Verdier商等局部化现象，刻画了局部化范畴为三角或正合范畴的等价条件，并应用于上同调函子相关结构的理解。

动机范畴的内部同伦理论

On the internal homotopy theory of motivic categories

该研究在∞-范畴框架下发展动机范畴的内部同伦理论，建立相关准则，证明双有理动机同伦范畴具有特定拓扑斯性质，为动机同伦范畴的性质提供关键结论。