A cryptic remark in Street--Walters, 'Yoneda structures on 2-categories'
Hello list, A theorem of Linton says that the Eilenberg--Moore category of a monad is equivalent to the category of presheaves on its Kleisli category that become representable when restricted to the base category (along the canonical inclusion). In Street and Walters's paper 'Yoneda structures on 2-categories', J. Algebra 50, 1978, proposition 22 generalizes Linton's result. The authors say in the introduction that
it shows that the Eilenberg--Moore algebras for a monad can be regarded as sheaves for a certain generalized topology on the Kleisli category.
Can anyone shed any light on what they mean by a `generalized topology'? FL -- Finn Lawler [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Finn, The sheaves for a topology can be defined as those presheaves which send certain diagrams to limits. The diagrams in question are determined by the topology. Similarly, the Elienberg-Moore algebras can be seen as the presheaves on the Kleisli category which send certain diagrams to limits. The limits in question, however, are not (usually) those determined by a topology, thus Street and Walters speak of a generalized topology. Regards, Steve Lack. On 11/01/2011, at 1:46 AM, Finn Lawler wrote:
Hello list,
A theorem of Linton says that the Eilenberg--Moore category of a monad is equivalent to the category of presheaves on its Kleisli category that become representable when restricted to the base category (along the canonical inclusion).
In Street and Walters's paper 'Yoneda structures on 2-categories', J. Algebra 50, 1978, proposition 22 generalizes Linton's result. The authors say in the introduction that
it shows that the Eilenberg--Moore algebras for a monad can be regarded as sheaves for a certain generalized topology on the Kleisli category.
Can anyone shed any light on what they mean by a `generalized topology'?
FL
-- Finn Lawler
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Finn On 11/01/2011, at 1:46 AM, Finn Lawler wrote:
it shows that the Eilenberg--Moore algebras for a monad can be regarded as sheaves for a certain generalized topology on the Kleisli category.
Can anyone shed any light on what they mean by a `generalized topology'?
Perhaps we should have said "generalized Ehresmann sketch". After all, I believe Linton's work aimed at generalizing to all monads the correspondence between monads of finite rank on Set and Lawvere theories, under which Eilenberg-Moore algebras become product-preserving presheaves on part of the Kleisli category). I suspect that is what we intended. Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Comment at end ... ------ Original Message ------ Received: Tue, 11 Jan 2011 07:42:45 AM EST From: Ross Street <ross.street@mq.edu.au> To: Finn Lawler <flawler@cs.tcd.ie> Subject: categories: Re: A cryptic remark in Street--Walters, 'Yoneda structures on 2-categories'
Dear Finn
On 11/01/2011, at 1:46 AM, Finn Lawler wrote:
it shows that the Eilenberg--Moore algebras for a monad can be regarded as sheaves for a certain generalized topology on the Kleisli category.
Can anyone shed any light on what they mean by a `generalized topology'?
Perhaps we should have said "generalized Ehresmann sketch".
After all, I believe Linton's work aimed at generalizing to all monads the correspondence between monads of finite rank on Set and Lawvere theories, under which Eilenberg-Moore algebras become product-preserving presheaves on part of the Kleisli category).
I suspect that is what we intended.
Ross
And actually even "product-preserving" is only a loose and not entirely appropriate slogan for the actual condition I promulgated in that work. I always thought that "generalized topology" was the result of caving in to a similar *manner-of-speaking* abbreviatory impulse, and I suppose today's "generalized Ehresmann sketch" is, as well. No harm done, of course ... :-) . Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
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Finn Lawler -
Fred E.J. Linton -
Ross Street -
Steve Lack