Jasso, Gustavo Higher Auslander algebras of type \(\mathbb{A}\) and the higher Waldhausen \(\mathsf{S}\)-constructions. (English) Zbl 1456.18011 Šťovíček, Jan (ed.) et al., Representation theory and beyond. Workshop and 18th international conference on representations of algebras, ICRA 2018, Prague, Czech Republic, August 13–17, 2018. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 758, 249-265 (2020). Summary: These notes are an expanded version of my talk at the ICRA 2018 in Prague, Czech Republic; they are based on joint work with T. Dyckerhoff et al. [Adv. Math. 355, Article ID 106762, 73 p. (2019; Zbl 1471.16027)]. In them we relate Iyama’s higher Auslander algebras of type \(\mathbb{A}\) to Eilenberg-Mac Lane spaces in algebraic topology and to higher-dimensional versions of the Waldhausen \(\mathsf{S}\)-construction from algebraic \(K\)-theory.For the entire collection see [Zbl 1461.16004]. MSC: 18G80 Derived categories, triangulated categories 18N50 Simplicial sets, simplicial objects 16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers 55U35 Abstract and axiomatic homotopy theory in algebraic topology Keywords:Auslander-Reiten theory; Eilenberg-Mac Lane spaces Citations:Zbl 1471.16027 PDFBibTeX XMLCite \textit{G. Jasso}, Contemp. Math. 758, 249--265 (2020; Zbl 1456.18011) Full Text: DOI arXiv References: [1] Angeleri H\"{u}gel, Lidia; Koenig, Steffen; Liu, Qunhua; Yang, Dong, Ladders and simplicity of derived module categories, J. Algebra, 472, 15-66 (2017) · Zbl 1373.16020 [2] Auslander, Maurice, Relations for Grothendieck groups of Artin algebras, Proc. Amer. Math. Soc., 91, 3, 336-340 (1984) · Zbl 0542.16028 [3] Be\u{\i}linson, A. A.; Bernstein, J.; Deligne, P., Faisceaux pervers. Analysis and topology on singular spaces, I, Luminy, 1981, Ast\'{e}risque 100, 5-171 (1982), Soc. Math. France, Paris [4] F. Beckert, The bivariant parasimplicial \(S_{}\)-construction, Ph.d. thesis, Bergische Universitat Wuppertal, July 2018. [5] Blumberg, Andrew J.; Gepner, David; Tabuada, Gon\c{c}alo, A universal characterization of higher algebraic \(K\)-theory, Geom. Topol., 17, 2, 733-838 (2013) · Zbl 1267.19001 [6] Bergh, Petter Andreas; Thaule, Marius, The Grothendieck group of an \(n\)-angulated category, J. Pure Appl. Algebra, 218, 2, 354-366 (2014) · Zbl 1291.18015 [7] Cisinski, Denis-Charles, Higher categories and homotopical algebra, Cambridge Studies in Advanced Mathematics 180, xviii+430 pp. (2019), Cambridge University Press, Cambridge · Zbl 1430.18001 [8] T. Dyckerhoff, G. Jasso, and T. Walde, Generalised BGP reflection functors via the Grothendieck construction, Int. Math. Res. Not. IMRN 11 (2019). DOI 10.1093/imrn/rnz194. [9] T. Dyckerhoff, G. Jasso, and T. Walde, Simplicial structures in higher Auslander-Reiten theory, Adv. Math. 355 (2019), 106762. \MR{3994443} · Zbl 1471.16027 [10] Dold, Albrecht, Homology of symmetric products and other functors of complexes, Ann. of Math. (2), 68, 54-80 (1958) · Zbl 0082.37701 [11] T. Dyckerhoff, A categorified Dold-Kan correspondence, arXiv:1710.08356 (2017). · Zbl 1468.18024 [12] Faonte, Giovanni, \( \mathcal{A}_\infty \)-functors and homotopy theory of dg-categories, J. Noncommut. Geom., 11, 3, 957-1000 (2017) · Zbl 1390.18034 [13] Faonte, Giovanni, Simplicial nerve of an \(\mathcal{A}_\infty \)-category, Theory Appl. Categ., 32, Paper No. 2, 31-52 (2017) · Zbl 1360.18023 [14] Goerss, Paul G.; Jardine, John F., Simplicial homotopy theory, Progress in Mathematics 174, xvi+510 pp. (1999), Birkh\"{a}user Verlag, Basel · Zbl 0949.55001 [15] M. Groth, A short course on \(\)-categories, arXiv:1007.2925 (2010). · Zbl 1473.55014 [16] Groth, Moritz; Stov\'{\i}\v{c}ek, Jan, Abstract representation theory of Dynkin quivers of type \(A\), Adv. Math., 293, 856-941 (2016) · Zbl 1345.55005 [17] Groth, Moritz; Stov\'{\i}\v{c}ek, Jan, Tilting theory for trees via stable homotopy theory, J. Pure Appl. Algebra, 220, 6, 2324-2363 (2016) · Zbl 1337.55024 [18] Groth, Moritz; Stov\'{\i}\v{c}ek, Jan, Tilting theory via stable homotopy theory, J. Reine Angew. Math., 743, 29-90 (2018) · Zbl 1403.18018 [19] Happel, Dieter, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series 119, x+208 pp. (1988), Cambridge University Press, Cambridge · Zbl 0635.16017 [20] L. Hesselholt and I. Madsen, Real algebraic \(K\)-theory, Unpublished, April 2015. [21] Hovey, Mark, Model categories, Mathematical Surveys and Monographs 63, xii+209 pp. (1999), American Mathematical Society, Providence, RI · Zbl 0909.55001 [22] Iyama, Osamu; Oppermann, Steffen, \(n\)-representation-finite algebras and \(n\)-APR tilting, Trans. Amer. Math. Soc., 363, 12, 6575-6614 (2011) · Zbl 1264.16015 [23] Iyama, Osamu, Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories, Adv. Math., 210, 1, 22-50 (2007) · Zbl 1115.16005 [24] Iyama, Osamu, Cluster tilting for higher Auslander algebras, Adv. Math., 226, 1, 1-61 (2011) · Zbl 1233.16014 [25] Jasso, Gustavo; K\"{u}lshammer, Julian, Higher Nakayama algebras I: Construction, Adv. Math., 351, 1139-1200 (2019) · Zbl 1427.16011 [26] Joyal, A., Quasi-categories and Kan complexes, J. Pure Appl. Algebra, 175, 1-3, 207-222 (2002) · Zbl 1015.18008 [27] Kan, Daniel M., Functors involving c.s.s. complexes, Trans. Amer. Math. Soc., 87, 330-346 (1958) · Zbl 0090.39001 [28] S. Ladkani, Homological properties of finite partially ordered sets, Ph.D. thesis, Hebrew University of Jerusalem, 2008. · Zbl 1127.18005 [29] Lurie, Jacob, Higher topos theory, Annals of Mathematics Studies 170, xviii+925 pp. (2009), Princeton University Press, Princeton, NJ · Zbl 1175.18001 [30] Jacob Lurie, Higher algebra, May 2017, Available online at http://www.math.harvard.edu/lurie/. · Zbl 1175.18001 [31] Oppermann, Steffen; Thomas, Hugh, Higher-dimensional cluster combinatorics and representation theory, J. Eur. Math. Soc. (JEMS), 14, 6, 1679-1737 (2012) · Zbl 1254.05197 [32] T. Poguntke, Higher Segal structures in algebraic \(K\)-theory, arXiv:1709.06510 (2017). [33] Quillen, Daniel G., Homotopical algebra, Lecture Notes in Mathematics, No. 43, iv+156 pp. (not consecutively paged) pp. (1967), Springer-Verlag, Berlin-New York [34] Waldhausen, Friedhelm, Algebraic \(K\)-theory of spaces. Algebraic and geometric topology, New Brunswick, N.J., 1983, Lecture Notes in Math. 1126, 318-419 (1985), Springer, Berlin This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.