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SUMMARY:Rune Haugseng (NTNU Trondheim)
DTSTART:20210204T100000Z
DTEND:20210204T113000Z
DTSTAMP:20260417T091740Z
UID:OWHHS2021/9
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OWHHS2021/9/
 ">Homotopy-coherent distributivity and the universal property of bispans</
 a>\nby Rune Haugseng (NTNU Trondheim) as part of Opening Workshop (IRP Hig
 her Homotopy Structures 2021\, CRM-Bellaterra)\n\n\nAbstract\nStructures w
 here we have both a contravariant (pullback) and a covariant (pushforward)
  functoriality that satisfy base change can be encoded by functors out of 
 ($\\infty$-)categories of spans (or correspondences). In some cases we hav
 e two pushforwards (an 'additive' and a 'multiplicative' one)\, satisfying
  a distributivity relation. Such structures can be described in terms of b
 ispans (or polynomial diagrams). For example\, commutative semirings can b
 e described in terms of bispans of finite sets\, while bispans in finite $
 G$-sets can be used to encode Tambara functors\, which are the structure o
 n $\\pi_0$ of $G$-equivariant commutative ring spectra. \n\nMotivated by a
 pplications of the ∞-categorical upgrade of such descriptions to motivic
  and equivariant ring spectra\, I will discuss the universal property of $
 (\\infty\,2)$-categories of bispans [1]. This gives a universal way to obt
 ain functors from bispans\, which amounts to upgrading 'monoid-like' struc
 tures to 'ring-like' ones. In the talk I will focus on the simplest case o
 f bispans in finite sets\, where this gives a new construction of the semi
 ring structure on a symmetric monoidal $\\infty$-category whose tensor pro
 duct commutes with coproducts. \n\nReferences:\n\n[1] Elden Elmanto and Ru
 ne Haugseng\, On distributivity in higher algebra I: The universal propert
 y of bispans\, arXiv:2010.15722.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/9/
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