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 H^{0} -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.