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The formal theory of theories

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If you have a question about this talk, please contact Nathanael Arkor.

Since Linton’s remarkable insight that the algebraic theories of Lawvere are equivalent to monads on the category of sets, many correspondences of a similar nature have been discovered, leading to increasingly general theorems relating notions of theory to classes of monads. Most recently, the work of Lucyshyn-Wright and of Bourke–Garner establishes tight monad–theory correspondences in the setting of enriched categories.

Despite the generality of these approaches, there are interesting examples that remain beyond reach, such as monads internal to topoi; graded monads; and Diers’s multimonads. More importantly, it is difficult to extract from the present approaches which assumptions are crucial to the monad–theory correspondence, and which arise simply as artefacts of the setting. Philosophically, we should like to know why the monad–theory correspondence holds, to the extent that it should appear an inevitable consequence of the definitions. These considerations motivate the study of a monad–theory correspondence at a greater level of abstraction.

In this talk, I will outline a purely formal perspective on the monad–theory correspondence, working in the setting of a 2-category with a suitable factorisation system and having enough Kleisli objects. The motivating examples 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 relative monads, the formal theory of which has recently been initiated by Lobbia. Though relative monads are a strict generalisation 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 correspondence. I shall pay particular attention to the monads 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 categories as a special case.

The first half of this talk concerns joint work with Dylan McDermott.

This talk is part of the Logic & Semantics for Dummies series.

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