Simplicial structures in higher Auslander-Reiten theory

Tobias Dyckerhoff, Gustavo Jasso, Tashi Walde

Research output: Contribution to journalArticlepeer-review


We develop a novel combinatorial perspective on the higher Auslander algebras of type A, a family of algebras arising in the context of Iyama's higher Auslander–Reiten theory. This approach reveals interesting simplicial structures hidden within the representation theory of these algebras and establishes direct connections to Eilenberg–MacLane spaces and higher-dimensional versions of Waldhausen's S-construction in algebraic K-theory. As an application of our techniques we provide a generalisation of the higher reflection functors of Iyama and Oppermann to representations with values in stable infinity-categories. The resulting combinatorial framework of slice mutation can be regarded as a higher-dimensional variant of the abstract representation theory of type quivers developed by Groth and Šťovíček. Our simplicial point of view then naturally leads to an interplay between slice mutation, horn filling conditions, and the higher Segal conditions of Dyckerhoff and Kapranov. In this context, we provide a classification of higher Segal objects with values in any abelian category or stable infinity-category.
Original languageEnglish
Article number106762
Number of pages73
JournalAdvances in Mathematics
Publication statusPublished - 2019
Externally publishedYes

Subject classification (UKÄ)

  • Algebra and Logic

Free keywords

  • Higher Auslander algebras
  • Higher Segal spaces
  • Tilting
  • Dold-Kan correspondence


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