Complicial simple-minded collections
Marvin Plogmann

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 -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.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Rings, Modules, and Algebras
