Re: flat topologies

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

Here's one way to look at it. A flat functor F: C --> Set corresponds to
a point p of the presheaf topos [C^op,Set]. Given a Grothendieck
topology J, p factors through the sheaf topos Sh(C,J) iff F carries
J-covering sieves to epimorphic families. Thus the particular J you
define is the largest for which p factors through Sh(C,J); equivalently,
Sh(C,J) is the image (in the surjection--inclusion sense) of the
geometric morphism p: Set --> [C^op,Set]. Hence a topos has a
presentation of this kind iff it admits a surjective geometric morphism
from Set; equivalently, iff it is (equivalent to) the category of
coalgebras for a finite-limit-preserving accessible comonad on Set.

Peter Johnstone
-------------------
On Fri, 10 Jul 2009, Szlachanyi Kornel wrote:

> Dear List,
>
> I have some questions on flat functors and Grothendieck topologies.
> It is probably well-known that if F: C--> Set is a flat functor
> on a small category then there is a Grothendieck topology on C
> in which the covering sieves S on the object c consist of sets
> of arrows to c such that {Fs|s in S} is jointly epimorphic on Fc.
>
> 1. Could you tell me a reference for this statement?
>
> 2. Is there a characterization of Grothendieck topologies that
> arise in this way from a flat functor? (`flat topologies'?)
>
> 3. For what categories C will there be a flat functor inducing
> the canonical topology on C?
>
> Thank you for any help.
>
> Kornel
>
> ---------------------------------------------------
> Kornel Szlachanyi
> Research Institute for Particle and Nuclear Physics
> of the Hungarian Academy of Science
> Budapest
> ---------------------------------------------------
>
>

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