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SUMMARY:Simona Paoli (University of Aberdeen)
DTSTART:20250703T091500Z
DTEND:20250703T101500Z
DTSTAMP:20260404T094546Z
UID:TTT/1
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/TTT/1
 /">A higher categorical approach to the André-Quillen cohomology of an (
 ∞\, 1)-Category</a>\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
BEGIN:VEVENT
SUMMARY:Nicholas Meadows (University of Bologna)
DTSTART:20250703T103000Z
DTEND:20250703T113000Z
DTSTAMP:20260404T094546Z
UID:TTT/2
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/TTT/2
 /">Andrè-Quillen Cohomology and the k-Invariants of Simplicial Categories
 </a>\nby Nicholas Meadows (University of Bologna) as part of Transalpine T
 opology Tetrahedron (TTT) - Pavia Vertex\n\nLecture held in Aula Beltrami.
 \n\nAbstract\nSpaces\, and more generally infinity-categories\, have a can
 onical decomposition into simpler pieces known as Postnikov sections\, whi
 ch are glued together by their k-invariants.  For an $\\infty$-category X\
 , these take value in the spectral Andre-Quillen cohomology of Harpaz\, Nu
 iten\, and Prasma. By pulling these k-invariants back to diagrams inside X
 \, one obtains a series of obstructions to lifting the diagram to successi
 ve stages in the Postnikov tower. \n\nIn this talk\, we will show how vari
 ous constructions in algebraic topology\, such as differentials in spectra
 l sequences and cohomology operations\, can be viewed as obstructions to e
 xtending cubical diagrams in the infinity category of spaces. Motivated by
  this\, we will also show that there exists a canonical cubical decomposit
 ion of the spectral Andre-Quillen cohomology. Joint work with David Blanc.
 \n\nJoin Zoom Meeting https://unipv-it.zoom.us/j/94344875868\n\nMeeting ID
 : 943 4487 5868\n
LOCATION:https://stable.researchseminars.org/talk/TTT/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luigi Caputi (University of Bologna)
DTSTART:20250703T140000Z
DTEND:20250703T150000Z
DTSTAMP:20260404T094546Z
UID:TTT/3
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/TTT/3
 /">The weak categorical quiver minor theorem and its applications</a>\nby 
 Luigi Caputi (University of Bologna) as part of Transalpine Topology Tetra
 hedron (TTT) - Pavia Vertex\n\nLecture held in Aula Beltrami.\n\nAbstract\
 nThe aim of the talk is to describe the weak categorical quiver minor theo
 rem. We will introduce the framework of quasi-Groebner categories\, as dev
 eloped 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 Miyata\, Proudfoot and Ramos. More specifically\, we will focus on mag
 nitude (co)homology\, as introduced by Hepworth and Willerton\, and we wil
 l show that magnitude cohomology yields finitely generated functors on the
  category of directed graphs with bounded genus. Then\, we will discuss so
 me main applications. This is joint work with Carlo Collari and Eric Ramos
 .\n\nJoin Zoom Meeting https://unipv-it.zoom.us/j/94344875868\n\nMeeting I
 D: 943 4487 5868\n
LOCATION:https://stable.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:20260404T094546Z
UID:TTT/4
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/TTT/4
 /">Motivic K-theory</a>\nby Paul Arne Østvær (University of Milano and U
 niversity of Oslo) as part of Transalpine Topology Tetrahedron (TTT) - Pav
 ia Vertex\n\nLecture held in Aula Beltrami.\n\nAbstract\nWe will discuss d
 efinitions and properties of motivic K-theory of motivic ring spectra. Joi
 nt work with Hadrian Heine.\n\nJoin Zoom Meeting https://unipv-it.zoom.us/
 j/94344875868\n\nMeeting ID: 943 4487 5868\n
LOCATION:https://stable.researchseminars.org/talk/TTT/4/
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