University of Birmingham > Talks@bham > Theoretical computer science seminar > From finitary monads to Lawvere theories: Cauchy completions

From finitary monads to Lawvere theories: Cauchy completions

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NB 217 not Sloman Lounge

The two main category theoretic formulations of universal algebra are Lawvere theories and finitary (= filtered-colimit preserving) monads on Set. The usual way in which to construct a Lawvere theory from a finitary monad T is by considering the opposite of the restriction of the Kleisli category Kl(T) to finite sets or equivalently natural numbers. Richard Garner recently found a different formulation of this, using the notion of Cauchy completion of a finitary monad qua monoid, i.e., qua one-object V-category, in V, where V is the monoidal category [Set_f,Set], equivalently the monoidal category of filtered- colimit preserving functors from Set to Set.

Both finitary monads (easily) and Lawvere theories (with more effort) extend from Set to arbitrary locally finitely presentable categories. So last year in Sydney, Richard and I explored the extension of his construction via Cauchy completions. That can only be done in a unified way, i.e., not for one locally finitely presentable category at a time, but for all simultaneously, using the notion of W-category for a bicategory W. I shall talk about as much of this as we can reasonably handle: it is work in progress, so I have not fully grasped it myself yet, and there is much to absorb, e.g, the concepts of Cauchy completion and categories enriched in bicategories. The emphasis will very likely be on Richard’s work rather than the work we did jointly.

(joint with Richard Garner)

This talk is part of the Theoretical computer science seminar series.

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