Let $F:\mathbf{\Delta}\to\mathcal{S}\_{\leqslant n-1}$ be a cosimplicial object in the $\infty$-category of $(n-1)$-truncated spaces. Is it always a right Kan extension of its restriction along $\mathbf{\Delta}^{\leqslant n}\hookrightarrow\mathbf{\Delta}$? (Then one might call $F$ to be $n$-coskeletal.)


I want to use this to show that *$\mathbf{\Delta}^{\leqslant n}\_{/[m]}:=\mathbf{\Delta}^{\leqslant n}\times\_{\mathbf{\Delta}}\mathbf{\Delta}\_{/[m]}$ is $n$-connected ($\pi\_{<n}\simeq 0$), for every $[m]\in\mathbf{\Delta}$* (which I view as a finite version of Quillen’s theorem A). Other ways to confirm is also fine (without solving the above question).


One might replace $\mathbf{\Delta}$ with $\mathbf{\Delta}\_s$ everywhere, e.g. by (proof of) [HTT, Lemma 6.5.3.8]; in the case $m\leqslant n$, it is easy to see that it has a final object hence is in fact contractible.



Connectedness of truncated version of cosimplicial indexing category
I don't believe the first claim is true, but I can give a somewhat formal argument for the connectivity of $\Delta\_{\le n} \times\_{\Delta} \Delta\_{/m}$.
Let us write $u$ for the inclusion $\Delta\_{\le n} \to \Delta$. The crucial result follows the following special case of Appendix A of <https://arxiv.org/abs/2207.09256>:


**Theorem 1.** Given a diagram $F : \Delta \to \mathcal{S}\_{\le n-1}$ the canonical map $\lim\_{\Delta} F \to \lim\_{\Delta\_{\le n}} u^\*F$ is invertible.


(This may already be the result you were after, but we can carry this through to actually obtain the connectivity result you requested.)


Fix an arbitrary $F : \Delta \to \mathcal{S}\_{\le n-1}$. By Theorem 1, we know that the following is an equivalence:


\begin{align\*}
&\lim F
\\
&\cong \hom\_{\mathbf{Fun}(\Delta,\mathcal{S})}(\mathbf{1}, F)
\\
&\to \hom\_{\mathbf{Fun}(\Delta\_{\le n},\mathcal{S})}(u^\*\mathbf{1}, u^\*F)
\\
&\cong \hom\_{\mathbf{Fun}(\Delta\_{\le n},\mathcal{S})}(\mathbf{1}, 
u^\*F)
\\
&\cong \lim u^\*F
\end{align\*}


Transposing, we conclude that $\hom\_{\mathbf{Fun}(\Delta,\mathcal{S})}(\mathbf{1}, F) \to \hom\_{\mathbf{Fun}(\Delta,\mathcal{S})}(u\_!u^\*\mathbf{1}, F)$ is an equivalence and, consequently, that $u\_!u^\* \mathbf{1} \to \mathbf{1}$ is orthogonal to any $(n-1)$-truncated object in $\mathbf{Fun}(\Delta, \mathcal{S})$. Accordingly, $u\_!u^\*\mathbf{1} = u\_!\mathbf{1}$ is $n$-connected. We therefore conclude that $(u\_!\mathbf{1})(m) : \mathcal{S}$ is $n$-connected for any $m : \Delta$. Let us unfold this using the formula for left Kan extensions:
$$
(u\_!\mathbf{1})(m) =
\mathrm{colim}\_{
 \Delta\_{\le n} 
 \times\_{\Delta}
 \Delta\_{/m}
} \mathbf{1}
= L(\Delta\_{\le n} \times\_{\Delta} \Delta\_{/m})
$$


The conclusion then follows.


Incidentally, this is just slight alteration of the usual proof of one direction of Quillen's theorem A. The other direction (that suitably connective fibers implies $n$-cofinality) also [holds](https://mathoverflow.net/questions/236578/difficulties-with-descent-data-as-homotopy-limit-of-image-of-%c4%8cech-nerve/236600#236600).


