Tuesday, March 15, 16:15

Xiaofa Chen (Paris / Hefei), What is an exact dg category?

Abstract:
In his recent preprint 'Exact DG-Categories' Positselski proposes an answer to the question in the title. In this talk, we will propose an entirely different answer based on the work of Barwick, who introduced exact ∞-categories in 2012 as an ∞-categorical generalization of Quillen's notion of exact category. We define exact dg categories in such a way that their dg nerves are exact ∞-categories in the sense of Barwick. In analogy with a theorem by Nakaoka-Palu (2020) we show that the H⁰ -category of an exact dg category carries a canonical extriangulated structure. We call such extriangulated categories algebraic, which extends the corresponding notion for triangulated categories. Typical examples are Yilin Wu's Higgs categories and Haibo Jin's categories of dg Cohen-Macaulay modules. We also show that each connective exact dg category embeds fully exactly into its dg derived category, in analogy with a theorem by Klemenc for exact ∞-categories. We expect that the connectivity assumption cannot be dispensed with in general.