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Re 2: In my opinion, none of the answers just yet have hit the nail on the head about "why this trick works". The real reason is Urs' comment about dense subsites. Whether one takes manifolds to mean …

Let us concentrate on Grothendieck topoi. As mentioned in earlier posts, these are those topoi which arise as the category of sheaves for a category equipped with a Grothendieck topology. These are th …

There is a full and faithful embedding of the category of schemes into the $2$-category of (edit: stricly Henselian ringed) topoi, which sends each scheme $X$ to the topos $Sh\left(X_{et}\right)$ (in …

This answer is not meant to discourage others from giving a complete answer, but only to help get towards a full one: (Thanks to Urs Schreiber for helping me work this out) For $C=Top$, you can prov …

Local epimorphisms are precisely those morphisms in $\mathcal{P}\left(\mathcal{C}\right)$ which become effective epimorphisms after applying the sheafification functor. In particular, if $f$ is an eff …

There is a canonical equivalence of $2$-categories $$St\left(Man/M\right) \simeq St\left(Man\right)/M$$ between stacks on the large site of $M$ and stacks on the site of manifolds equipped with a ma …

This not a complete answer but too long for a comment: First a quick remark: 1.) and 4.) generate the same topology. Now let geometric realization be denoted by $G$. Consider the geometric morphism …