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Let $T, P$ be two topoi, and $f:T \longrightarrow P$.

Does there exist two site $S_{T}, S_{P}$ and a morphism $g: S_{T} \longrightarrow S_{P}$ such that $f$ is induced by $g$ ?

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Yes: every morphism of Grothendieck toposes arises as a morphism of sites. (However, the site may depend on the morphism.) This is Corollary C2.3.10 in Johnstone's Sketches of an elephant. The argument goes something like this: pick any site $\mathcal{S}_\mathcal{P}$ for the codomain $\mathcal{P}$; then we may take as $\mathcal{S}_\mathcal{T}$ some subcategory of $\mathcal{T}$ that contains a site for $\mathcal{T}$ as well as the images under $f^*$ of all representable sheaves on $\mathcal{S}_\mathcal{P}$.

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  • $\begingroup$ Not every topos is a Grothendieck topos, and for non Grothendieck toposes the answer is: no, such sites do not exist. $\endgroup$ – Wouter Stekelenburg Apr 3 '13 at 6:39
  • $\begingroup$ Indeed, but that is almost by definition. $\endgroup$ – Zhen Lin Apr 3 '13 at 7:05

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