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Given an infinity topos $\mathcal{E}$, one can pass to its hypercompletion $\widehat{\mathcal{E}}$, which is the full subcategory on those objects local with respect to $\infty$-connected morphisms (morphisms inducing isomorphisms on all homotopy sheaves). Can passing to the hypercompletion be made functorial with respect to all geometric morphisms? (For my purposes, it would suffice to know functoriality with respect to etale ones). Thanks!

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    $\begingroup$ According to 6.5.2.13 in Higher Topos Theory, hypercompletion is right adjoint to the inclusion of hypercomplete $\infty$-topoi into all $\infty$-topoi. Doesn't that make it functorial? $\endgroup$ – Mike Shulman Mar 1 '13 at 2:48
  • $\begingroup$ @Mike: 6.5.2.13 would imply right-adjointness, once we know it's functorial. That's kind of why I asked. $\endgroup$ – David Carchedi Mar 1 '13 at 10:53
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    $\begingroup$ In 1-category theory, a universal property of objects like that expressed by 6.5.2.13 implies a unique way to extend the operation to a functor that is a right adjoint. See Theorem IV.1.2 in "Categories for the working mathematician". I presume the same must be true for $(\infty,1)$-categories. $\endgroup$ – Mike Shulman Mar 1 '13 at 11:05
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    $\begingroup$ @Mike: Good point, I guess I was foggy in the morning, thanks :). $\endgroup$ – David Carchedi Mar 1 '13 at 20:05

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