LFCS seminar: John Power: From finitary monads to Lawvere theories: Cauchy completions
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Aug 15, 2017 from 04:00 PM to 05:00 PM 
Where  IF 4.31/4.33 
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(joint with Richard Garner)
The
two main category theoretic formulations of universal algebra are
Lawvere theories and finitary (= filteredcolimit 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 oneobject Vcategory, in V, where V is the monoidal category
[Set_f,Set], equivalently the monoidal category [Set,Set]f of
filteredcolimit 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 works most naturally if one does it in a
unified way, i.e., not for one locally finitely presentable category at a
time, but for all simultaneously, using the notion of Wcategory 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.