The simplicial $n$-sphere is the unique simplicial set with one non-degenerate 0-simplex, one non-degenerate $n$-simplex, and no other non-degenerate simplices. The $n$-sphere is obtained from $\Delta^n$ by identifying the boundary to a point.
We define reduced as a simplicial set having on one element in $K_0$.
So, we know that the simplicial $n$-sphere is a reduced simplicial set. However, is the simplicial $n$-sphere a Kan complex?
Specifically, I'm looking at the 2-sphere. So, the nondegenerate simplices are $[0]\in X_0$ and $[0,1,2]\in X_2$.
One way to define a simplicial set $X$ as a Kan complex is if it satisfies the following Kan condition: The simplicial set $X$ satisifies the Kan condition if for any collection of $(n-1)$-simplices $x_0,...,x_{k-1},x_{k+1},...,x_n$ in $X$ such that $d_ix_j = d_{j-1}x_i$ for any $i<j$ with $i\neq k$ and $j\neq k$, there is an $n$-simplex $x$ in $X$ such that $d_ix = x_i$ for all $i\neq k$.
Note: Here $d_i$ are face maps.
(The condition on the simplicies $x_i$ of this definition glues them together to form the horn $\Lambda_{k}^{n}$, possibly with degenerate faces, within $X$, and the definition says that we can extend this horn to a (possibly degenerate) $n$-simplex in $X$. )
In our case the only $(n-1)$-simplex or $1$-simplex would be the degenerate simplex $[0,0]\in X_1$.
So, would this be a Kan complex trivially because we only have one $(n-1)$-simplex for the 2-sphere? So, we satisfy the condition trivially.
Or rather, since the only 1-simplex is $[0,0]\in X_1$ and the only 2-simplex is $[0,1,2]$ and we don't satisfy the condition $d_ix=x_i$ (since $d_0[0,1,2] = [1,2]\neq [0,0]$, $d_1[0,1,2] = [0,2]\neq [0,0]$, and $d_2[0,1,2] = [0,1]\neq [0,0]$), then the 2-sphere is not a Kan complex?