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SUMMARY:Free monads in double categories - Nicola Gambino\, University of 
 Palermo
DTSTART:20101102T141500Z
DTEND:20101102T151500Z
UID:TALK27551@talks.cam.ac.uk
CONTACT:Nathan Bowler
DESCRIPTION:The development of the formal theory of monads\, begun by\nStr
 eet and later continued by Street and Lack\, shows that large parts of the
  theory of monads can be developed within an arbitrary 2-category rather t
 han in the 2-category of small categories\, functors and natural transform
 ations. I will describe some joint work with Tom Fiore and Joachim Kock in
  which we extend the basic concepts of the formal theory of monads from th
 e setting of 2-categories to that of double categories. The motivation to 
 do so derives from the desire to understand better the universal propertie
 s of the free category on a graph and of the free monad on a polynomial en
 dofunctor. Our main result shows that\, under some mild conditions\, a dou
 ble category that\nis a framed bicategory admits the construction of free 
 monads if its horizontal 2-category does. After explaining this result\, I
  will illustrate how it can be applied to obtain double adjunctions that e
 xtend the adjunction between graphs and categories and the adjunction betw
 een polynomial endofunctors and polynomial monads.
LOCATION:MR3\, Centre for Mathematical Sciences
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