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Thanks for your comment. You are right that I should have asked about $\mathbb{R}Hom(A,BG_m)$ instead, I have edited (hopefully this make sense even if $BG_m$ is not derived). Perhaps I should have just asked if there was a well defined notion of derived dual abelian scheme.
More generally, a functor $F$ between accessible categories is a left (resp. right) adjoint iff it is accessible and preserves all small colimits (resp. limits). This is in Borceux's book. If $C$ is locally presentable, and $F$ preserves colimits, then it is automatically accessible (because $C$ is cocomplete, or alternatively because it is total).
@David: Indeed, imposing the diagonal to be representable indeed feels more "geometric". But in this case the equivalence with the groupoid approach fails no? (Or rather we then need to impose the diagonal condition everywhere.) For the terminology, I used the one from the nlab: ncatlab.org/nlab/show/geometric+stack. But I have also seen the presentable stack terminology (maybe from your articles?).
@Marc: wait, if $(C,τ)$ is affine schemes with the étale topology, then geometric stacks are stacks with an affine étale (representable) presentation, and they do correspond to étale affine groupoids and étendues (since $F$ corresponds to $U \to X$ which is étale, and representable since pullbacks are above $F$ so should be affine by definition of an étendue).
Also the saturation condition reminds me of a comment I forgot to make on requiring $\tau$-covering to satisfy the descent condition: it looks like the key condition to bound the number of time we iterate the construction before it saturates; something like iterating n+2 time to get n-geometric stacks. We can even troncate along the way, so iterate twice to get geometric sheafs, then geometric 1-stack, 2-stacks and so on (so for algebraic stack this would go affine -> alg space with affine diagonal -> alg space -> alg stack rather than the more standard affine->schemes->alg space...).
So in this case the link with an étendue is less clear. Still for general algebraic stack, we can see them as stacks for the fppf topology with an fppf presentation, or as stacks for the smooth topology (hence the étale topology) with smooth presentation. So I guess the equivalence with étendues still holds (but the only proof I know is from Pronk and she only treat the DM case).
Yes, this is precisely the type of things that make me wish for an exhaustive treatment of when the equivalence holds. Your comment show two things: 1) we want to have some kind of saturation condition on $C$, eg a geometric stack that is a sheaf should be in $C$. 2) In my assumptions I use the same type of morphisms for the topology and for presentations. So for the étale topology, we require a geometric stack to have an étale presentation, so to be DM. A more exhaustive treatment would distinguish between the topology and the presentations (like in HAG2).
So I think we mostly agree that unramified functors simplify the deformation-theoretic arguments, but my point of view is that more generally the smooth case simplify deformation-theoretic arguments. To get the unramified case we just need the fact that a sheaf which is étale locally an algebraic space is algebraic.
Well formal versality is rather easy to check using Schlessinger's criterion. Then you need algebraicity (for effectivity). Artin's approximation is a big theorem, but can be used as a black box. In practice the hard part to check is openness of versality. But for a smooth alg space (over the base), this is a lot easier to check (no obstruction to deformations). In particular this apply in the étale case. Now an unramified morphism is étale locally a closed immersion. So essentially you just need the étale case of Artin's axioms, and then use representability of loc. qf separated morphisms.
