Re: (In)accessible comonads and (non)Grothendieck toposes

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

A particular case of this is in SGA4, IV 9.5.4, where it is shown
that the category obtained by (Artin) glueing along a finite-limit-
preserving functor between Grothendieck toposes is a Grothendieck
topos iff the functor is accessible. It was Gavin Wraith, in JPAA 4
(1974), who first observed that Artin glueing is a particular case
of forming a category of coalgebras (and therefore works for
elementary toposes without the accessibility condition).

Peter Johnstone

On Thu, 9 May 2013, David Roberts wrote:

> Hi all,
>
> I am just wondering where it was first stated (for both directions) that 
> the category of coalgebras for a comonad on a Grothendieck topos E is 
> again Grothendieck if and only if the underlying endofunctor of E is 
> accessible.
>
> A modern argument might go as: the topos of coalgebras is Grothendieck 
> if and only if it is locally presentable if and only if the endofunctor 
> is accessible, the original probably just mentioned preservation of 
> filtered colimits.
>
> Many thanks,
>
> David Roberts
>
>


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