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SUMMARY:Simona Paoli (University of Aberdeen)
DTSTART:20250703T091500Z
DTEND:20250703T101500Z
DTSTAMP:20260423T005718Z
UID:TTT/1
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/TTT/1/">A hi
 gher categorical approach to the André-Quillen cohomology of an (∞\, 1)
 -Category</a>\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/
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