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SUMMARY:Containers\, Comonads and Distributive Laws - Danel Ahman
DTSTART:20120625T114500Z
DTEND:20120625T130000Z
UID:TALK38619@talks.cam.ac.uk
CONTACT:Peter Sewell
DESCRIPTION:Joint work with James Chapman and Tarmo Uustalu from Institute
  of Cybernetics\, Tallinn\n\n\nAbbott\, Altenkirch\, Ghani and others have
  taught us that many parameterized datatypes can\nbe usefully analyzed via
  container representations in terms of shapes and positions.\n\nOur work b
 uilds on the observation that datatypes often carry additional structure t
 hat containers alone\ndo not account for. We introduce directed containers
  to capture the common situation where every position\nin a datastructure 
 determines another datastructure\, informally\, the sub-datastructure root
 ed by that position.\nSome natural examples are non-empty lists and node-l
 abelled trees\, and datastructures with a designated position\n(zippers).\
 n\nWhile containers denote set functors via a fully-faithful functor\, dir
 ected containers interpret fully-faithfully\ninto comonads. But more is tr
 ue: every comonad whose underlying functor is a container is represented b
 y a\ndirected container. In fact\, directed containers are the same as con
 tainers that are comonads.\n\nSimilarly to comonads\, directed containers 
 do not generally compose. However\, a sufficient condition for two\ncomona
 ds to compose is the existence of a distributive law between them. We deve
 lop a corresponding theory for\ndirected containers\, present the distribu
 tive-law based composition of two directed containers and show that it\nge
 neralizes the Zappa-Sz\\'ep product of two monoids.
LOCATION:FW26
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