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SUMMARY:Timothy Campion (Johns Hopkins University)
DTSTART:20230925T070000Z
DTEND:20230925T083000Z
DTSTAMP:20260422T161058Z
UID:HKUST-AG/19
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/HKUST-AG/19/
 ">Smooth and proper algebras via stable $(\\infty\,2)$-categories</a>\nby 
 Timothy Campion (Johns Hopkins University) as part of Algebra and Geometry
  Seminar @ HKUST\n\nLecture held in Room 5560.\n\nAbstract\nSince Grothend
 ieck\, the notion of an abelian 1-category has provided a natural setting 
 to do algebra which encompasses both categories of modules and categories 
 of sheaves. Since Lurie\, the notion of a stable $(\\infty\,1)$-category h
 as provided a similar setting to do derived algebra\, encompassing derived
  categories of modules and sheaves\, and improving upon the notion of a tr
 iangulated category due to Verdier.\n\nIn this talk\, we discuss a few pos
 sible notions of stable $(\\infty\,2)$-category\, motivated by enriched ca
 tegory theory. Examples include the $(\\infty\,2)$-category of dg categori
 es\, the $(\\infty\,2)$-category of stable $(\\infty\,1)$-categories\, and
  various $(\\infty\,2)$-categories of stacks of stable $(\\infty\,1)$-cate
 gories. The intention is to provide a natural home for the study of such $
 (\\infty\,2)$-categories\, which are of interest in areas such as the Geom
 etric Langlands program\, secondary algebraic K-theory\, and derived algeb
 raic geometry.\n\nWe discuss work in progress on showing that our notions 
 of stable $(\\infty\,2)$-category are equivalent. As an application\, we s
 how for example that every smooth and proper algebra over a regular commut
 ative Noetherian ring k may be constructed from $k$ by iterating two simpl
 e operations: glueing along a perfect bimodule\, and 2-idempotent splittin
 g.\n
LOCATION:https://researchseminars.org/talk/HKUST-AG/19/
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