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:20260422T212726Z UID:TTT/1 DESCRIPTION:Title: A hi gher categorical approach to the André-Quillen cohomology of an (∞\, 1) -Category\nby Simona Paoli (University of Aberdeen) as part of Transal pine Topology Tetrahedron (TTT) - Pavia Vertex\n\nLecture held in Aula Bel trami.\n\nAbstract\nSimplicial categories\, that is categories enriched in simplicial sets\, are a model of (∞\, 1)-categories. Their André-Quill en cohomology\, originally introduced by Dwyer\, Kan and Smith [DKS]\, was later re-interpreted and extended by Harpaz\, Nuiten and Prasma [HNP1]. T he André-Quillen cohomology of a simplicial category can be used to descr ibe its k-invariants which in turn contain various higher homotopy informa tion and in particular yield an obstruction theory for realizing homotopy- commutative diagrams [DKS]. Our aim is to give an algebraic\, elementary a nd explicit approach to the André-Quillen cohomology of simplicial catego ries using the tools of higher category theory.\n\n For this purpose\, we first observe that in order to study the nth André-Quillen cohomology gro up of a simplicial category\, it suffices to look at simplicial categories 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 hig her category theory to produce an algebraic replacement for the nth Postni kov truncation of a simplicial category. We choose to use the category 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 with a car tesian monoidal structure. Further\, every n-type can be modelled by a wea kly globular n-fold groupoid\, that is an object of the full subcategory o f weakly globular n-fold groupoids [BP2]\, which is more convenient algebr aically. Our model for the nth Postnikov truncation of a simplicial catego ry is a category enriched in weakly globular n-fold groupoids with respect to the cartesian monoidal structure. We call the latter an n-track catego ry. Using the n-fold nature of our model\, we iteratively build a comonad on n-track categories. Using this comonad we then obtain an explicit cosim plicial abelian group model for the André-Quillen cohomology of an (∞\, 1)-category. This is joint work with David Blanc [BP4].\n\n\nReferences:\ n\n[BP2] D. Blanc & S. Paoli\, Segal-type algebraic models of n-types\, Al gebraic & Geometric Topology 14 (2014)\, pp. 3419-3491.\n\n[BP4] D. Blanc & S. Paoli\, A Model for the André-Quillen Cohomology of an (∞\, 1)-Cat egory\, preprint arXiv:2405.12674v2\, 2024.\n\n[DKS] W.G. Dwyer\, D.M. Kan \, J. H. Smith An obstruction theory for diagrams of simplicial categories \, Proc.Kon. Ned. Akad. Wet. - Ind. Math. 48 (1986)\, pp. 153-161. \n\n[HN P1] Y. Harpaz\, J. Nuiten\, & M. Prasma\, The abstract cotangent complex a nd Quillen cohomology of enriched categories\, J. Topology 11 (2018)\, 752 -798.\n\n[Pa3] S. Paoli\, Simplicial Methods for Higher Categories: Segal- type models of weak n-categories\, 'Algebra and Applications'\, Springer\, Berlin-New York\, 2019.\n\nJoin Zoom Meeting https://unipv-it.zoom.us/j/9 4344875868\nMeeting ID: 943 4487 5868\n LOCATION:https://researchseminars.org/talk/TTT/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Nicholas Meadows (University of Bologna) DTSTART:20250703T103000Z DTEND:20250703T113000Z DTSTAMP:20260422T212726Z UID:TTT/2 DESCRIPTION:Title: Andr è-Quillen Cohomology and the k-Invariants of Simplicial Categories\nb y Nicholas Meadows (University of Bologna) as part of Transalpine Topology Tetrahedron (TTT) - Pavia Vertex\n\nLecture held in Aula Beltrami.\n\nAbs tract\nSpaces\, and more generally infinity-categories\, have a canonical decomposition into simpler pieces known as Postnikov sections\, which are glued together by their k-invariants. For an $\\infty$-category X\, these take value in the spectral Andre-Quillen cohomology of Harpaz\, Nuiten\, and Prasma. By pulling these k-invariants back to diagrams inside X\, one obtains a series of obstructions to lifting the diagram to successive stag es in the Postnikov tower. \n\nIn this talk\, we will show how various con structions in algebraic topology\, such as differentials in spectral seque nces and cohomology operations\, can be viewed as obstructions to extendin g cubical diagrams in the infinity category of spaces. Motivated by this\, we will also show that there exists a canonical cubical decomposition of the spectral Andre-Quillen cohomology. Joint work with David Blanc.\n\nJoi n Zoom Meeting https://unipv-it.zoom.us/j/94344875868\n\nMeeting ID: 943 4 487 5868\n LOCATION:https://researchseminars.org/talk/TTT/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Luigi Caputi (University of Bologna) DTSTART:20250703T140000Z DTEND:20250703T150000Z DTSTAMP:20260422T212726Z UID:TTT/3 DESCRIPTION:Title: The weak categorical quiver minor theorem and its applications\nby Luigi C aputi (University of Bologna) as part of Transalpine Topology Tetrahedron (TTT) - Pavia Vertex\n\nLecture held in Aula Beltrami.\n\nAbstract\nThe ai m of the talk is to describe the weak categorical quiver minor theorem. We will introduce the framework of quasi-Groebner categories\, as developed by Sam and Snowden\, and use it to study structural properties (e.g. bound on ranks and order of torsion) of graph homologies\, in the spirit of Miy ata\, Proudfoot and Ramos. More specifically\, we will focus on magnitude (co)homology\, as introduced by Hepworth and Willerton\, and we will show that magnitude cohomology yields finitely generated functors on the catego ry of directed graphs with bounded genus. Then\, we will discuss some main applications. This is joint work with Carlo Collari and Eric Ramos.\n\nJo in Zoom Meeting https://unipv-it.zoom.us/j/94344875868\n\nMeeting ID: 943 4487 5868\n LOCATION:https://researchseminars.org/talk/TTT/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Paul Arne Østvær (University of Milano and University of Oslo) DTSTART:20250703T153000Z DTEND:20250703T163000Z DTSTAMP:20260422T212726Z UID:TTT/4 DESCRIPTION:Title: Moti vic K-theory\nby Paul Arne Østvær (University of Milano and Universi ty of Oslo) as part of Transalpine Topology Tetrahedron (TTT) - Pavia Vert ex\n\nLecture held in Aula Beltrami.\n\nAbstract\nWe will discuss definiti ons and properties of motivic K-theory of motivic ring spectra. Joint work with Hadrian Heine.\n\nJoin Zoom Meeting https://unipv-it.zoom.us/j/94344 875868\n\nMeeting ID: 943 4487 5868\n LOCATION:https://researchseminars.org/talk/TTT/4/ END:VEVENT END:VCALENDAR