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 higherdimensional 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 ∞-categories. The resulting combinatorial framework of slice mutation can be regarded as a higher-dimensional variant of the abstract representation theory of type A quivers developed by Groth andŠťovíček. 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 ∞-category.
Inspired by work of Ladkani and Groth–Šťovíček, we explain how to construct generalisations of the classical reflection functors of Bernšteĭn, Gel′fand, and Ponomarev by means of the Grothendieck construction.
We show that the various higher Segal conditions of Dyckerhoff and Kapranov can all be characterized in purely categorical terms by higher excision conditions (in the spirit of Goodwillie-Weiss manifold calculus) on the simplex category ∆ and the cyclic category Λ.
We study complexes of stable 8-categories, referred to as categorical complexes. As we demonstrate, examples of such complexes arise in a variety of subjects including representation theory, algebraic geometry, symplectic geometry, and differential topology. One of the key techniques we introduce is a totalization construction for categorical cubes which is particularly well-behaved in the presence of Beck-Chevalley conditions. As a direct application we establish a categorical Koszul duality result which generalizes previously known derived Morita equivalences among higher Auslander algebras and puts them into a conceptual context. We explain how spherical categorical complexes can be interpreted as higher-dimensional perverse schobers, and introduce Calabi-Yau structures on categorical complexes to capture noncommutative orientation data. A variant of homological mirror symmetry for categorical complexes is proposed and verified for CP 2 . Finally, we develop the concept of a lax additive p8, 2q-category and propose it as a suitable framework to formulate further aspects of categorified homological algebra.
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