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I'm working out of Sheaves in geometry and logic, for reference.

There is a characterisation of flat functors $A:C \to Set$ as those such that the Grothendieck construction $\int_C A$ is a filtering category. There are more general versions of this result, in which $Set$ is replaced by a more general topos. One should also be able to characterise those discrete opfibrations that arise from flat functors (up to iso/equiv?). How about if we replace $C$ by an internal category, in a topos $E$ say? Then functors out of $C$ are replaced by discrete opfibrations over $C$ in $E$.

My question is this:

What sort of thing should be considered as the analogue of a flat functor in the internal setting?

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A flat internal presheaf/discrete fibration $F \to C$ is simply one whose total category F is filtered in the internal sense (see Topos Theory 2.51 (filteredness), 4.31 (flatness); Elephant B.2.6.2, B.3.2.3). The definition just reexpresses filteredness by requiring that certain maps (e.g. $F_0 \to 1$) are regular epis (presumably you can replace these with covers for your favourite Grothendieck topology).

I think that if C = BG is a group then these are exactly the G-torsors in E. I'm not sure whether this extends to flat = locally representable for C a general internal category, though.

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    $\begingroup$ Just to clarify, the Topos Theory reference here is to Johnstone's 1977 book. Iirc it also has a couple of good exercises on this topic. $\endgroup$
    – Peter LeFanu Lumsdaine
    Aug 20, 2010 at 15:42

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