On this nlab page a functor $\mathbf{cosk_n}:sSet\to sSet$ is constructed.
Is $\mathbf{cosk_n}(K)$ of a Kan complex $K$ again a Kan complex?
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$\mathbf{cosk}_n$ does indeed preserve Kan complexes.
Let $\mathbf{sk}_n$ be the left adjoint of $\mathbf{cosk}_n$. Given a horn inclusion $\Lambda^m_k \hookrightarrow \Delta^m$, the morphism $\mathbf{sk}_n (\Lambda^m_k \hookrightarrow \Delta^m)$ is an isomorphism if $n > m + 1$, so by adjointness, $\mathbf{cosk}_n (K)$ has the right lifting property with respect to $\Lambda^m_k \hookrightarrow \Delta^m$ for all $n > m + 1$, whether or not $K$ is a Kan complex.
On the other hand, $\mathbf{sk}_n (\Lambda^m_k \hookrightarrow \Delta^m)$ is (canonically isomorphic to) $\Lambda^m_k \hookrightarrow \Delta^m$ itself when $m \le n$, so $\mathbf{cosk}_n (K)$ has the right lifting property with respect to $\Lambda^m_k \hookrightarrow \Delta^m$ if and only if $K$ does, when $m \le n$.
That leaves $m = n + 1$. Here, $\mathbf{sk}_n (\Lambda^{n+1}_k \hookrightarrow \Delta^{n+1})$ is isomorphic to $\Lambda^{n+1}_k \hookrightarrow \partial \Delta^{n+1}$, so $\mathbf{cosk}_n (K)$ has the right lifting property with respect to $\Lambda^{n+1}_k \hookrightarrow \Delta^{n+1}$ if and only if $K$ has the right lifting property with respect to $\Lambda^{n+1}_k \hookrightarrow \partial \Delta^{n+1}$. This is not an anodyne extension, but Kan complexes nonetheless have the right lifting property with respect to it: indeed, choose a horn filler as usual and restrict along the inclusion $\partial \Delta^{n+1} \hookrightarrow \Delta^{n+1}$.
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$\begingroup$ Dear Zhen Lin, do you know if the coskeleton functor also preserves Kan fibrations between Kan complexes? I am in particular interested in $cosk_0(-)$. Thank you. $\endgroup$ – user8463524 Apr 21 '15 at 8:54
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2$\begingroup$ If you examine the argument above, you will see that $\mathbf{cosk}_n$ preserves Kan fibrations whose codomain has the right lifting property with respect to the inclusion $\partial \Delta^{n+1} \hookrightarrow \Delta^{n+1}$. For a Kan complex, this is equivalent to the condition that $\pi_n$ is trivial. In particular, $\mathbf{cosk}_0$ preserves Kan fibrations whose codomain is a connected Kan complex. $\endgroup$ – Zhen Lin Apr 21 '15 at 9:10
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$\begingroup$ I think that in the second paragraph, the two $n>m+1$ conditions should be $m>n+1.$ $\endgroup$ – Dap Apr 9 at 16:27
