01-06【Bernhard Keller】管楼1208 吴文俊数学重点实验室代数学系列报告之196


题目:On extriangulated categories, exact infinity-categories and exact dg categories (I, II, III)

报告人:Bernhard Keller 教授,巴黎大学

时间:2021年12月17日下午15:30-17:30     地点:东区管理科研楼1318教室

          2021年12月24日下午15:30-17:30     地点:东区管理科研楼1318教室

          2022年1月6日上午11:00 开始                 地点:东区管理科研楼1208教室

报告人简介: Bernhard Keller是著名代数学家,巴黎大学教授, 中国科学技术大学客座教授, 在微分分次理论、丛理论以及Hochschild同调理论中均做出了奠基性的学术成果。Keller教授是挪威皇家科学通讯院士、比利时安特卫普大学荣誉博士、法国科学院Sophie Germain奖得主、国际数学家大会ICM邀请报告人以及美国数学会Fellow,任国际知名杂志Advances in Mathematics以及Forum of Mathematics Pi 编委。


In this lecture series, we will present recent developments concerning the three generalizations of Quillen's notion of exact category mentioned in the title.

In the first lecture, we will introduce extriangulated categories following the work of Nakaoka-Palu (2019). This notion generalizes both, the notion of exact and that of triangulated category. It arises naturally in the categorification of cluster algebras with coefficients. Here the relevant categories are due to Pressland (for many examples) and Yilin Wu (in full generality).

The second lecture will be devoted to infinity-categories (modeled using  quasi-categories)and more specifically to exact infinity-categories in the sense of  Barwick (2015 and 2016).The link to extriangulated categories is given by Nakaoka-Palu's theorem  (04/2020) stating that the homotopy category of an exact infinity-category carries a  canonical extriangulated structure. Such extriangulated categories are called *topological* extriangulated categories.

The third lecture is motivated by the search for a suitable notion of  *algebraic* extriangulated category, i.e. a class of differential graded (=dg) k-categories whose  H^0 carries a natural extriangulated structure in analogy of with that of topological  extriangulated categories.We will present the solution proposed by Xiaofa Chen in his ongoing Ph. D thesis. It seems very likely that Lurie's dg nerve functor transforms an exact dg category in the sense of Chen into an exact infinity-category in the sense of Barwick and that  the exact infinity-categories obtained in this way are precisely those admitting a k-linear structure.