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Suppose that we have two Grothendieck sites, their associated sheaves $\mathcal{E}=\rm{Sh}(\bf{C},J)$ and $\mathcal{F}=\rm{Sh}(\bf{D},K)$ and a geometric surjection $f:\mathcal{E}\to\mathcal{F}$. This is completely determined by a faithful, left-exact, cover-preserving functor $f_0:\bf{D}\to \mathcal{E}$, which induces the inverse image $f^*$ by left Kan extension.

We should be able to play the same game for stacks. Any stack $G$ over $(\bf{D},K)$ has a presentation as a (pseudo)colimit of representables: $G\simeq\varinjlim_i yD_i$ and we set $f^*G\simeq\varinjlim_i f(D_i)$. Does this induce a "2-geometric morphism" $\rm{Stacks}(\bf{C},J)\to\rm{Stacks}(\bf{D},K)$. In what sense, if any, is this morphism a "2-surjection"?

In the case of sheaves, the geometric morphism induces a left-exact comonad on $\mathcal{E}$ and $\mathcal{F}$ is the category of coalgebras for this comonad. Is there an analogous statement for stacks?

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I'll try to come back and answer this later when I have more time (it's late here), but for now: since you get a morphism between the toposes of sheaves on the two sites, you get a morphism between their 2-toposes of stacks, since there is a full and faithful embedding of the bicategory of toposes into the the tricategory of 2-toposes. –  David Carchedi Apr 23 '13 at 23:27
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