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Suppose we fix two Grothendieck universes $\mathcal{U} \in \mathcal{V}.$ Then one has that $\mathcal{U}$-$\mathbf{Set},$ the category of $\mathcal{U}$-small sets, is a locally $\mathcal{U}$-small, $\mathcal{V}$-small category. Grothendieck universes were used often by, well, Grothendieck, in his work with topoi. Part of the reason was to work with "large sites". One important example is that for a given topos $\mathcal{E},$ it carries its canonical topology. Grothendieck universes can be used to make sense of the statement that $\mathcal{E}$ is equivalent to sheaves over itself with respect to this topology, but lets look at another idea. If $\mathcal{E}=Sh\left(C,J\right)$ is a sheaf topos, we will instead of considering sheaves with values in the large category of all sets, consider only sheaves with values in $\mathcal{U}$-$\mathbf{Set}.$ And this is a $\mathcal{U}$-topos. Lets take this idea and run with it:

There is a functor from $\mathcal{U}$-topoi to $\mathcal{V}$-topoi which sends a $\mathcal{U}$-topos $\mathcal{E}$ to $$Sh\left(\mathcal{E},\mathcal{V}\mbox{-}\mathbf{Set}\right),$$ the $\mathcal{V}$-topos of sheaves on $\mathcal{E}$ with its canonical topology, considered as a $\mathcal{V}$-small site.

This easily generalizes for $n$-topoi for any $n$. Are these functors full and faithful?

For $n=0$ the answer is yes, and remains yes for $n=1$ if we restrict to localic topoi, for topological reasons- but is it true for higher $n$?

(I foresee a possible complication for $\infty$-topoi which are not equivalent to sheaves on a site.)

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See related question mathoverflow.net/questions/24552/… for various strictly weaker universe concepts that can substitute for universes in many arguments and constructions. –  Joel David Hamkins Oct 1 '12 at 23:56
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Perhaps it would be more fruitful to ask about the model-theoretic relationship between the two topoi (i.e., whether the truth of some class of first-order sentences is preserved) instead of the categorical relationship. –  S. Carnahan Oct 2 '12 at 7:53
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up vote 16 down vote accepted

The change-of-universe construction is faithful but not full. For example, let X be the topos of sets and let Y be the classifying topos for abelian groups. The category of geometric morphisms from X to Y is equivalent to the category of abelian groups. If you pass to a larger universe, you get more abelian groups.

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So simple and clear, thanks! –  David Carchedi Oct 2 '12 at 10:42
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I guess it is only full for $n=0$ since $(-1)$-groupoids are always just the set of truth values, true and false, regardless of universe, but for any higher $n$, there are more $n$-groupoids in a bigger universe. –  David Carchedi Oct 2 '12 at 17:40
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