### The bare question:


Let $\mathcal{C}$ be an $\infty$-topos, and let $\tau\_{\leq 0}\mathcal{C}$ be the subcategory of 0-truncated objects (which is the nerve of an ordinary Grothendieck topos: see HTT 6.4.1.3).



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> Does the inclusion $\tau\_{\leq 0}\mathcal{C} \hookrightarrow \mathcal{C}$ preserve filtered (or at least directed) ($\infty$-)colimits?
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> 
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### Motivation:


Let $Aff\_\mathbb{C}$ the Grothendieck site of complex affine schemes. We can then consider the topos of sheaves, $Shv(Aff\_\mathbb{C})$, and the $\infty$-topos of $\infty$-stacks, $Shv\_\infty(Aff\_\mathbb{C})$. The nerve of the first is equivalent to the subcategory of 0-truncated objects in the second.


Given a scheme $X$ and a closed subscheme $Y$ in it defined by a sheaf of ideals $\mathcal{I}$, we can construct the so-called *formal completion of $X$ along $Y$* as the directed colimit $X\_Y^{\mbox{^}} = \mathrm{colim}\:V(\mathcal{I}^n)$. Typically this is done in $Shv(Aff\_\mathbb{C})$. Since the homotopy theory in the latter is trivial, it is also the homotopy colimit of the same diagram. But how does this play with the inclusion $Shv(Aff\_\mathbb{C}) \hookrightarrow Shv\_\infty(Aff\_\mathbb{C})$? Is $X\_Y^{\mbox{^}}$ still the homotopy colimit of the same diagram in $Shv\_\infty(Aff\_\mathbb{C})$?


An affirmative answer to my last question would be enough for me, but I suppose it is a natural question to ask whether this holds for general formal schemes —i.e., sheaves that are locally formal completions as above.



FIltered colimits of truncated objects in $\infty$-topoi
I believe the answer is YES and, more generally, that $\tau\_{\leq n}\mathcal{C}\subset\mathcal{C}$ preserves filtered colimits for any $\infty$-topos $\mathcal{C}$. For the $\infty$-topos of $\infty$-groupoids this is well-known. This implies the result in any presheaf $\infty$-topos since colimits and truncations are computed objectwise. Finally, if the result is true in $\mathcal{C}$ and $\mathcal{D}\subset\mathcal{C}$ is a left exact localization, then the result is true in $\mathcal{D}$ as well because left exact functors preserve $n$-truncated objects (HTT, Prop. 5.5.6.16).


