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Exact dg categories I : Foundations

Xiaofa Chen

Abstract

We introduce the notion of exact dg category, which provides a differential graded enhancement of Nakaoka--Palu's notion of extriangulated category. We give a definition in complete analogy with Quillen's but where the category of kernel-cokernel pairs is replaced with a more sophisticated homotopy category. We introduce the notion of stable dg category, and prove that the $H^0$-category of an exact dg category $\mathcal{A}$ is triangulated if and only if $\mathcal{A}$ is stable. We illustrate our theory with several examples including the homotopy category of two-term complexes and Amiot's fundamental domain for generalized cluster categories.

Exact dg categories I : Foundations

Abstract

We introduce the notion of exact dg category, which provides a differential graded enhancement of Nakaoka--Palu's notion of extriangulated category. We give a definition in complete analogy with Quillen's but where the category of kernel-cokernel pairs is replaced with a more sophisticated homotopy category. We introduce the notion of stable dg category, and prove that the -category of an exact dg category is triangulated if and only if is stable. We illustrate our theory with several examples including the homotopy category of two-term complexes and Amiot's fundamental domain for generalized cluster categories.
Paper Structure (21 sections, 48 theorems, 188 equations)

This paper contains 21 sections, 48 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Recollections on dg categories, ${A}_{\infty}$-categories and homotopy diagrams
  3. Dg categories
  4. ${A}_{\infty}$-categories
  5. Homotopy diagrams
  6. Homotopy pullbacks and homotopy pushouts
  7. Homotopy pullbacks and homotopy pushouts
  8. An $A_{\infty}$-theoretic description
  9. Some diagram lemmas
  10. Restriction of morphisms in $\mathcal{H}_{3t}(\mathcal{A})$
  11. Homotopy pushouts/pullbacks of homotopy short exact sequences
  12. Exact dg categories
  13. Exact dg structure
  14. The biadditive bifunctor $\mathbb E$
  15. The canonical extriangulated structure on $H^0(\mathcal{A})$
  16. ...and 6 more sections

Key Result

Lemma 2.2

The adjunction \begin{tikzcd} \dgcat_{\leq 0}\ar[r,"i", shift left =0.8ex]&\dgcat,\ar[l,"\tau_{\leq 0}", shift left=0.8ex] \end{tikzcd}where the left adjoint $i:\mathrm{dgcat}_{\leq 0}\rightarrow \mathrm{dgcat}$ is the inclusion functor and the right adjoint $\tau_{\leq 0}:\mathrm{dgcat}\rightarrow

Theorems & Definitions (128)

  • Definition : Lemma \ref{['lem:3termhcomplex']}
  • Definition : Definition \ref{['def:stable']}
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • ...and 118 more