###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).

Does the inclusion $\tau_{\leq 0}\mathcal{C} \hookrightarrow \mathcal{C}$ preserve filtered (or at least directed) ($\infty$-)colimits?

###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.

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).

Does the inclusion $\tau_{\leq 0}\mathcal{C} \hookrightarrow \mathcal{C}$ preserve filtered (or at least directed) ($\infty$-)colimits?

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.