BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Simona Paoli (University of Aberdeen) DTSTART:20250703T091500Z DTEND:20250703T101500Z DTSTAMP:20260404T131147Z UID:TTT/1 DESCRIPTION:Title: A higher categorical approach to the André-Quillen cohomology of an ( ∞\, 1)-Category\nby Simona Paoli (University of Aberdeen) as part of Transalpine Topology Tetrahedron (TTT) - Pavia Vertex\n\nLecture held in Aula Beltrami.\n\nAbstract\nSimplicial categories\, that is categories enr iched in simplicial sets\, are a model of (∞\, 1)-categories. Their Andr é-Quillen cohomology\, originally introduced by Dwyer\, Kan and Smith [DK S]\, was later re-interpreted and extended by Harpaz\, Nuiten and Prasma [ HNP1]. The André-Quillen cohomology of a simplicial category can be used to describe its k-invariants which in turn contain various higher homotopy information and in particular yield an obstruction theory for realizing h omotopy-commutative diagrams [DKS]. Our aim is to give an algebraic\, elem entary and explicit approach to the André-Quillen cohomology of simplicia l categories using the tools of higher category theory.\n\n For this purpo se\, we first observe that in order to study the nth André-Quillen cohomo logy group of a simplicial category\, it suffices to look at simplicial ca tegories that are n-truncated\, that is they are enriched in n-types. This has the advantage that we can use one of the algebraic models of n-types from higher category theory to produce an algebraic replacement for the nt h Postnikov truncation of a simplicial category. We choose to use the cate gory of groupoidal weakly globular n-fold categories arising within Paoli' s model of weak n-categories [Pa3]. This category is a model of n-types wi th a cartesian monoidal structure. Further\, every n-type can be modelled by a weakly globular n-fold groupoid\, that is an object of the full subca tegory of weakly globular n-fold groupoids [BP2]\, which is more convenien t algebraically. Our model for the nth Postnikov truncation of a simplicia l category is a category enriched in weakly globular n-fold groupoids with respect to the cartesian monoidal structure. We call the latter an n-trac k category. Using the n-fold nature of our model\, we iteratively build a comonad on n-track categories. Using this comonad we then obtain an explic it cosimplicial abelian group model for the André-Quillen cohomology of a n (∞\, 1)-category. This is joint work with David Blanc [BP4].\n\n\nRefe rences:\n\n[BP2] D. Blanc & S. Paoli\, Segal-type algebraic models of n-ty pes\, Algebraic & Geometric Topology 14 (2014)\, pp. 3419-3491.\n\n[BP4] D . Blanc & S. Paoli\, A Model for the André-Quillen Cohomology of an (∞\ , 1)-Category\, preprint arXiv:2405.12674v2\, 2024.\n\n[DKS] W.G. Dwyer\, D.M. Kan\, J. H. Smith An obstruction theory for diagrams of simplicial ca tegories\, Proc.Kon. Ned. Akad. Wet. - Ind. Math. 48 (1986)\, pp. 153-161. \n\n[HNP1] Y. Harpaz\, J. Nuiten\, & M. Prasma\, The abstract cotangent c omplex and Quillen cohomology of enriched categories\, J. Topology 11 (201 8)\, 752-798.\n\n[Pa3] S. Paoli\, Simplicial Methods for Higher Categories : Segal-type models of weak n-categories\, 'Algebra and Applications'\, Sp ringer\, Berlin-New York\, 2019.\n\nJoin Zoom Meeting https://unipv-it.zoo m.us/j/94344875868\nMeeting ID: 943 4487 5868\n LOCATION:https://stable.researchseminars.org/talk/TTT/1/ END:VEVENT END:VCALENDAR