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/ ]