Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Complicial simple-minded collections

Marvin Plogmann

Abstract

We consider the problem of characterizing derived endomorphism algebras of simple objects in length categories up to quasi-isomorphism. We give such a characterization for module categories, abelian categories, exact categories, as well as, for certain differential graded analogues of them. It turns out that the property of being $d$-complicial ($d\geq 1$), in the sense of Lurie, of the involved simple-minded collections plays a central role. We also explain how this characterization can be interpreted as a coherent generation property for any minimal $A_{\infty}$-model of the derived endomorphism algebra. Along the way, we propose a notion of length exact differential graded categories and explain how they relate to length abelian $d$-truncated differential graded categories, generalizing results of Enomoto.

Complicial simple-minded collections

Abstract

We consider the problem of characterizing derived endomorphism algebras of simple objects in length categories up to quasi-isomorphism. We give such a characterization for module categories, abelian categories, exact categories, as well as, for certain differential graded analogues of them. It turns out that the property of being -complicial (), in the sense of Lurie, of the involved simple-minded collections plays a central role. We also explain how this characterization can be interpreted as a coherent generation property for any minimal -model of the derived endomorphism algebra. Along the way, we propose a notion of length exact differential graded categories and explain how they relate to length abelian -truncated differential graded categories, generalizing results of Enomoto.
Paper Structure (19 sections, 49 theorems, 155 equations)

This paper contains 19 sections, 49 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Context and relation to other work
  3. Preliminaries
  4. Terminology and notation
  5. $t$-Structures, simple-minded collections and silting collections
  6. $A_{\infty}$-algebras
  7. Exact dg categories
  8. $\delta$-functors
  9. Dg realization functors
  10. Dg realization functors for exact dg subcategories
  11. Realization functors for extended hearts of $t$-structures
  12. Koszul duality for dg algebras
  13. Koszul duality for locally-finite connective dg algebras
  14. Koszul duality for proper connective dg algebras
  15. Coherently-generated $A_{\infty}$-algebras
  16. ...and 4 more sections

Key Result

Theorem 1.2

Let $k$ be a perfect field and $d\in\mathbb{N}_{>0}$. Koszul duality yields a bijective correspondence between quasi-isomorphism classes of the following two classes of dg algebras:

Theorems & Definitions (120)

  • Definition 1.1
  • Theorem 1.2: \ref{['theorem:KoszulDperfect']}
  • Theorem 1.3: \ref{['GeneralisedRealFunctor']}
  • Definition 1.4
  • Theorem 1.5: \ref{['compliciality']}
  • Theorem 1.6: \ref{['theorem:Enomoto']}
  • Corollary 1.7: \ref{['coro:AbelianSchur']}
  • Theorem 1.8: \ref{['theorem:SummaryAbelian']}
  • Theorem 1.9: \ref{['KoszulExact']}
  • Definition 2.1
  • ...and 110 more