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SUMMARY:Richard Garner (Macquarie University)
DTSTART:20200811T001500Z
DTEND:20200811T011500Z
DTSTAMP:20260423T004136Z
UID:operad/1
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/operad/1/">S
 weedler duals for monads</a>\nby Richard Garner (Macquarie University) as 
 part of operad pop-up\n\n\nAbstract\nWhile the linear dual of any coalgebr
 a is an algebra\, the converse is not true\; however\, there is an adjoint
  to the coalgebra-to-algebra functor\, given by the so-called Sweedler dua
 l.\n\nThere is a notion of “linear dual” for an endofunctor of Set\, g
 iven by homming into the identity functor for the Day convolution structur
 e. Again\, this sends comonads to monads\, but not vice versa\; but again\
 , there is an adjoint. This “Sweedler dual” comonad of a monad was int
 roduced by Katsumata\, Rivas and Uustalu in 2019.\n\nThe purpose of this t
 alk is to give an explicit construction of the Sweedler dual comonad of an
 y monad on Set. The category of coalgebras for the Sweedler dual turns out
  to be a presheaf category\, whose indexing category can be described expl
 icitly in terms of a kind of computational dynamics of the monad. If time 
 permits\, we also describe the source-etale topological category which cla
 ssifies the topological Sweedler dual comonad of a monad on Set\; in parti
 cular\, this recovers all kinds of etale topological groupoids of interest
  in the study of combinatorial $C^*$-algebras.\n
LOCATION:https://researchseminars.org/talk/operad/1/
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