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CATEGORIES:Semantics Lunch (Computer Laboratory)
SUMMARY:Containers\, Comonads and Distributive Laws - Dane
l Ahman
DTSTART;TZID=Europe/London:20120625T124500
DTEND;TZID=Europe/London:20120625T140000
UID:TALK38619AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/38619
DESCRIPTION:Joint work with James Chapman and Tarmo Uustalu fr
om Institute of Cybernetics\, Tallinn\n\n\nAbbott\
, Altenkirch\, Ghani and others have taught us tha
t many parameterized datatypes can\nbe usefully an
alyzed via container representations in terms of s
hapes and positions.\n\nOur work builds on the obs
ervation that datatypes often carry additional str
ucture that containers alone\ndo not account for.
We introduce directed containers to capture the co
mmon situation where every position\nin a datastru
cture determines another datastructure\, informall
y\, the sub-datastructure rooted by that position.
\nSome natural examples are non-empty lists and no
de-labelled trees\, and datastructures with a desi
gnated position\n(zippers).\n\nWhile containers de
note set functors via a fully-faithful functor\, d
irected containers interpret fully-faithfully\nint
o comonads. But more is true: every comonad whose
underlying functor is a container is represented b
y a\ndirected container. In fact\, directed contai
ners are the same as containers that are comonads.
\n\nSimilarly to comonads\, directed containers do
not generally compose. However\, a sufficient con
dition for two\ncomonads to compose is the existen
ce of a distributive law between them. We develop
a corresponding theory for\ndirected containers\,
present the distributive-law based composition of
two directed containers and show that it\ngenerali
zes the Zappa-Sz\\'ep product of two monoids.
LOCATION:FW26
CONTACT:Peter Sewell
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