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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