cases for $n = 2$. $$\Lambda^0[2] = \xymatrix{
 &1\ar@{ ..}[dr]^{\leftarrow 0^{th} \mbox{~\scriptsize{face missing}}}&\\
                0\ar[ur]\ar[rr]&&2}$$ $$\Lambda^1[2] = \xymatrix{
 &1\ar[dr]^{\hspace{2.3cm}}&\\
                0\ar[ur]\ar@{ ..}[rr]&&2}$$ $$\Lambda^2[2] = \xymatrix{
 &1\ar[dr]^{\hspace{2.3cm}}&\\
                0\ar@{ ..}[ur]\ar[rr]&&2}$$ A map $p: E \rightarrow B$ is a *Kan fibration* if given any $n$, $i$ as above and any $(n,i)$-horn in $E$, i.e. any map $f_1 : \Lambda^i[n]
\rightarrow E$, and $n$-simplex, $f_0 : \Delta[n] \rightarrow B$, such that $$\xymatrix{\Lambda^i[n]\ar[r]^{f_1}\ar[d]_{inc}& E\ar[d]^p\\
\Delta[n]\ar[r]_{f_0}&B}$$ commutes, then there is an $f : \Delta[n] \rightarrow E$ such that $pf = f_0$ and $f.inc = f_1$, i.e. $f$ lifts $f_0$ and extends $f_1$.

A simplicial set, $K$, is a *Kan complex* if the unique map $K
\rightarrow \Delta[0]$ is a Kan fibration. This is equivalent to saying that every horn in $K$ has a filler, i.e. any $f_1 : \Lambda^i[n]\rightarrow Y$ extends to an $f : \Delta[n] \rightarrow Y$. This condition looks to be purely of a geometric nature but in fact has an important algebraic flavour; for instance if $f_1 : \Lambda^1[2]\rightarrow Y$ is a horn, it consists of a diagram $$\xymatrix{ & \ar[dr]^b&\\\ar[ur]^a&&}$$ of ‘composable’ arrows in $K$. If $f$ is a filler, it looks like $$\xymatrix{ & \ar[dr]^b&\\ \ar[rr]_c\ar[ur]^a_{\quad f}&&}$$ and one can think of the 