Simplicial structures in higher Auslander-Reiten theory

Abstract

We develop a novel combinatorial perspective on the higher Auslander algebras of type $\mathbb{A}$, 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 $S_{\bullet }$-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 $\infty$-categories. The resulting combinatorial framework of slice mutation can be regarded as a higher-dimensional variant of the abstract representation theory of type $\mathbb{A}$ quivers developed by Groth and Šťov\'ıček. 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 $\infty$-category.

52 pages; v2: corrected minor mistake in Notation 1.12 and other typos

References in corpus (15)

• Higher Topos Theory
• Auslander correspondence
• $n$-angulated categories
• Higher Segal spaces I
• $n$-abelian and $n$-exact categories
• Triangulated surfaces in triangulated categories
• n-representation-finite algebras and twisted fractionally Calabi-Yau algebras
• Higher Nakayama algebras I: Construction
• Higher Segal structures in algebraic $K$-theory
• Abstract representation theory of Dynkin quivers of type A
• A1-homotopy invariants of topological Fukaya categories of surfaces
• Representation theory of Geigle-Lenzing complete intersections
• Spectra for commutative algebraists
• The Grothendieck group of an n-angulated category
• Abstract cubical homotopy theory

Cited by in corpus (9)

• Higher Nakayama algebras I: Construction
• The symplectic geometry of higher Auslander algebras: Symmetric products of disks
• A characterisation of higher torsion classes
• Coherent cochain complexes and Beilinson t-structures, with an appendix by Achim Krause
• Higher Segal spaces and Lax ${\mathbb{A}}_{\infty }$-algebras
• Higher Auslander algebras of type $\mathbb{A}$ and the higher Waldhausen $S$-constructions
• 3-Preprojective Algebras of Type D
• Higher symmetries in abstract stable homotopy theories
• The bivariant parasimplicial ${\mathsf{S}}_{\bullet }$-construction