Re: A cryptic remark in Street--Walters, 'Yoneda structures on 2-categories'

Re: A cryptic remark in Street--Walters, 'Yoneda structures on 2-categories'

[Fred E.J. Linton]

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



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Re: A cryptic remark in Street--Walters, 'Yoneda structures on 2-categories'

[Steve Lack]

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/ ]



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A cryptic remark in Street--Walters, 'Yoneda structures on 2-categories'

[Finn Lawler]

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


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