BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//talks.cam.ac.uk//v3//EN
BEGIN:VTIMEZONE
TZID:Europe/London
BEGIN:DAYLIGHT
TZOFFSETFROM:+0000
TZOFFSETTO:+0100
TZNAME:BST
DTSTART:19700329T010000
RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU
END:DAYLIGHT
BEGIN:STANDARD
TZOFFSETFROM:+0100
TZOFFSETTO:+0000
TZNAME:GMT
DTSTART:19701025T020000
RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
CATEGORIES:Logic &\; Semantics for Dummies
SUMMARY:The formal theory of theories - Nathanael Arkor (U
niversity of Cambridge)
DTSTART;TZID=Europe/London:20210827T150000
DTEND;TZID=Europe/London:20210827T160000
UID:TALK161647AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/161647
DESCRIPTION:Since Linton's remarkable insight that the algebra
ic theories of Lawvere are equivalent to monads on
the category of sets\, many correspondences of a
similar nature have been discovered\, leading to i
ncreasingly general theorems relating notions of t
heory to classes of monads. Most recently\, the wo
rk of Lucyshyn-Wright and of Bourke–Garner establi
shes tight monad–theory correspondences in the set
ting of enriched categories.\n\nDespite the genera
lity of these approaches\, there are interesting e
xamples that remain beyond reach\, such as monads
internal to topoi\; graded monads\; and Diers's mu
ltimonads. More importantly\, it is difficult to e
xtract from the present approaches which assumptio
ns are crucial to the monad–theory correspondence\
, and which arise simply as artefacts of the setti
ng. Philosophically\, we should like to know _why_
the monad–theory correspondence holds\, to the ex
tent that it should appear an inevitable consequen
ce of the definitions. These considerations motiva
te the study of a monad–theory correspondence at a
greater level of abstraction.\n\nIn this talk\, I
will outline a purely formal perspective on the m
onad–theory correspondence\, working in the settin
g of a 2-category with a suitable factorisation sy
stem and having enough Kleisli objects. The motiva
ting examples are given by the proarrow equipments
of Wood admitting finite tight collages (that is\
, those satisfying Wood's Axioms 4 and 5). It prov
es to be edifying first to establish a corresponde
nce between theories and _relative_ monads\, the f
ormal theory of which has recently been initiated
by Lobbia. Though relative monads are a strict gen
eralisation of monads\, there are circumstances in
which a relative monad may be extended to a monad
with the same Eilenberg–Moore object\, and it is
this situation that yields a monad–theory correspo
ndence. I shall pay particular attention to the mo
nads relative to the unit of a KZ doctrine\, which
form an important class of examples. In practice\
, most monad–theory correspondences arise in this
manner\, and we recover the setting of enriched ca
tegories as a special case.\n\nThe first half of t
his talk concerns joint work with Dylan McDermott.
LOCATION:https://meet.google.com/jxy-edcv-wgx
CONTACT:Nathanael Arkor
END:VEVENT
END:VCALENDAR