Simplicial structures in higher Auslander–Reiten theory

Dyckerhoff, Tobias; Jasso, Gustavo; Walde, Tashi

doi:10.1016/j.aim.2019.106762

Cited by 19 publications

(24 citation statements)

References 36 publications

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“…For > 1, as discussed for abelian categories in the work of Poguntke [Pog17] and in [Dyc17b,DJW19] in the stable context, the sequence perf( , ) of perfect derived (DG-)categories also forms a simplicial category. The relation to Waldhausen -theory is then given by the formula…”

Section: Thusmentioning

confidence: 99%

The symplectic geometry of higher Auslander algebras: Symmetric products of disks

Dyckerhoff

Jasso

Lekili

2021

Forum of Mathematics, Sigma

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We show that the perfect derived categories of Iyama’s d-dimensional Auslander algebras of type ${\mathbb {A}}$ are equivalent to the partially wrapped Fukaya categories of the d-fold symmetric product of the $2$ -dimensional unit disk with finitely many stops on its boundary. Furthermore, we observe that Koszul duality provides an equivalence between the partially wrapped Fukaya categories associated to the d-fold symmetric product of the disk and those of its $(n-d)$ -fold symmetric product; this observation leads to a symplectic proof of a theorem of Beckert concerning the derived Morita equivalence between the corresponding higher Auslander algebras of type ${\mathbb {A}}$ . As a by-product of our results, we deduce that the partially wrapped Fukaya categories associated to the d-fold symmetric product of the disk organise into a paracyclic object equivalent to the d-dimensional Waldhausen $\text {S}_{\bullet }$ -construction, a simplicial space whose geometric realisation provides the d-fold delooping of the connective algebraic K-theory space of the ring of coefficients.

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Section: Thusmentioning

confidence: 99%

The symplectic geometry of higher Auslander algebras: Symmetric products of disks

Dyckerhoff

Jasso

Lekili

2021

Forum of Mathematics, Sigma

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“…In the rest of the section, we will make free use of some results about cubic diagrams in stable ∞-categories, as presented in [DJW19], and refer to op. cit.…”

Section: Filtered Spectra and Cochain Complexes Of Spectramentioning

confidence: 99%

“…for all the related concepts, notations and terminology we use and do not introduce here. The following fact is somewhat implicit in [DJW19], but as we will use it crucially, we report it here for convenience.…”

Section: Filtered Spectra and Cochain Complexes Of Spectramentioning

confidence: 99%

Coherent cochain complexes and Beilinson t-structures, with an appendix by Achim Krause

Ariotta

2021

Preprint

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We define and study coherent cochain complexes in arbitrary stable ∞-categories, following Joyal. Our main result is that the ∞-category of coherent cochain complexes in a stable ∞-category C is equivalent to the ∞-category of complete filtered objects in C.We then show how the Beilinson t-structure can be interpreted in light of such equivalence, and analyze its behavior in the presence of symmetric monoidal structures. We also examine the relationship between the notion of (higher) Toda brackets and coherent cochain complexes. Finally, we prove how every coherent cochain complex gives rise to a spectral sequence and illustrate some examples.

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“…Indeed, triangulations of cyclic polytopes are often used to define higher-dimensional analogues of structures that exist for lower-dimensional triangulations. The example of this par excellence is the application of triangulations of cyclic polytopes to define higher Segal spaces [DK19], see also [Pog17;DJW19]. In integrable systems, regular triangulations of cyclic polytopes describe the evolution of a class of solitary waves modelled by the Kadomtsev-Petviashvili equation [DM12;Wil20b], see also [Hua15;KK21;GPW19].…”

Section: Introductionmentioning

confidence: 99%

The two higher Stasheff-Tamari orders are equal

Williams¹

2021

Preprint

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The set of triangulations of a cyclic polytope possesses two a priori different partial orders, known as the higher Stasheff-Tamari orders. The first of these orders was introduced by Kapranov and Voevodsky, while the second order was introduced by Edelman and Reiner, who also conjectured the two to coincide in 1996. In this paper we prove their conjecture, thereby substantially increasing our understanding of these orders. This result also has ramifications in the representation theory of algebras, as established in previous work of the author. Indeed, it means that the two corresponding orders on tilting modules, cluster-tilting objects and their maximal chains are equal for the higher Auslander algebras of type A.

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Simplicial structures in higher Auslander–Reiten theory

Cited by 19 publications

References 36 publications

The symplectic geometry of higher Auslander algebras: Symmetric products of disks

The symplectic geometry of higher Auslander algebras: Symmetric products of disks

Coherent cochain complexes and Beilinson t-structures, with an appendix by Achim Krause

The two higher Stasheff-Tamari orders are equal

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