Complicial simple-minded collections

TL;DR

This paper characterizes derived endomorphism algebras of simple objects in various length categories, emphasizing the role of $d$-complicial simple-minded collections and their relation to minimal $A_{ abla}$-models.

Contribution

It provides a new characterization of derived endomorphism algebras using $d$-complicial simple-minded collections across multiple categorical contexts.

Findings

Characterization of derived endomorphism algebras in module, abelian, and exact categories.

Introduction of length exact differential graded categories.

Connection between $d$-complicial collections and minimal $A_{ abla}$-models.

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.