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SUMMARY:The formal theory of theories - Nathanael Arkor (University of Cam
 bridge)
DTSTART:20210827T140000Z
DTEND:20210827T150000Z
UID:TALK161647@talks.cam.ac.uk
CONTACT:Nathanael Arkor
DESCRIPTION:Since Linton's remarkable insight that the algebraic theories 
 of Lawvere are equivalent to monads on the category of sets\, many corresp
 ondences of a similar nature have been discovered\, leading to increasingl
 y general theorems relating notions of theory to classes of monads. Most r
 ecently\, the work of Lucyshyn-Wright and of Bourke–Garner establishes t
 ight monad–theory correspondences in the setting of enriched categories.
 \n\nDespite the generality of these approaches\, there are interesting exa
 mples that remain beyond reach\, such as monads internal to topoi\; graded
  monads\; and Diers's multimonads. More importantly\, it is difficult to e
 xtract from the present approaches which assumptions are crucial to the mo
 nad–theory correspondence\, and which arise simply as artefacts of the s
 etting. Philosophically\, we should like to know _why_ the monad–theory 
 correspondence holds\, to the extent that it should appear an inevitable c
 onsequence of the definitions. These considerations motivate the study of 
 a monad–theory correspondence at a greater level of abstraction.\n\nIn t
 his talk\, I will outline a purely formal perspective on the monad–theor
 y correspondence\, working in the setting of a 2-category with a suitable 
 factorisation system and having enough Kleisli objects. The motivating exa
 mples are given by the proarrow equipments of Wood admitting finite tight 
 collages (that is\, those satisfying Wood's Axioms 4 and 5). It proves to 
 be edifying first to establish a correspondence between theories and _rela
 tive_ monads\, the formal theory of which has recently been initiated by L
 obbia. Though relative monads are a strict generalisation of monads\, ther
 e are circumstances in which a relative monad may be extended to a monad w
 ith the same Eilenberg–Moore object\, and it is this situation that yiel
 ds a monad–theory correspondence. I shall pay particular attention to th
 e monads relative to the unit of a KZ doctrine\, which form an important c
 lass of examples. In practice\, most monad–theory correspondences arise 
 in this manner\, and we recover the setting of enriched categories as a sp
 ecial case.\n\nThe first half of this talk concerns joint work with Dylan 
 McDermott.
LOCATION:https://meet.google.com/jxy-edcv-wgx
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