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CATEGORIES:Theoretical computer science seminar
SUMMARY:Towards a Calculus of Substitution for Dinatural T
ransformations - Alessio Santamaria (University of
Sussex)
DTSTART:20230505T130000Z
DTEND:20230505T135000Z
UID:TALK5245AT
URL:/talk/index/5245
DESCRIPTION:*Zoom details*\n\n* Link: https://bham-ac-uk.zoom.
us/j/81873335084?pwd=T1NaUFg2U1l6d0RLL2RlTzFBam1IU
T09\n* Meeting ID: 818 7333 5084\n* Passcode: 217\
n\n*Abstract*\n\nIn 1972\, with the papers "Many-V
ariable Functorial Calculus\, I" and "An Abstract
Approach of Coherence"\, Kelly started a long-term
project on achieving an abstract theory of cohere
nce. To do so\, he argues that a "tidy calculus of
substitution" of functors in many variables and a
ppropriately general natural transformations is in
order\; such a calculus ought to generalise the u
sual Godement calculus of functors in one variable
and ordinary natural transformations. Inspired by
the work he did with Eilenberg on extranatural tr
ansformations in 1966\, he developed this calculus
for (many-variable) covariant functors and natura
l transformations.\nUpon trying the mixed-variance
case\, he ran into problems linked to the fact th
at extranaturals do not compose. In this talk\, I
will show how we realised that the full mixed-vari
ance case wanted by Kelly involves a simple genera
lisation of dinatural transformations\; a sufficie
nt and essentially necessary condition for two con
secutive dinatural transformations to compose will
be mentioned\, and I will present a new definitio
n of horizontal composition of dinaturals. Armed w
ith these results\, I will show how to achieve he
first steps made by Kelly (in the mixed-variance c
ase\, this time) towards a full substitution calcu
lus. There are still some conceptual difficulties
about the remaining steps which are yet to be over
come.
LOCATION:LG23\, Computer Science // Zoom
CONTACT:George Kaye
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