Using Finn's observation that it is the subobject classifier, we can see that it is the nerve of the contractible groupoid with exactly two objects, $G_2$.

To see this, notice that $L_0=\{0,1\}$ induced by the two subobjects of $\Delta^0$, namely the empty map and the identity map.  It has four 1-cells induced by the empty map, the inclusion of the first vertex, the inclusion of the second vertex, and the identity.  It has eight 2-cells, etc.

The nerve of $G_2$ has two $0$-cells, four $1-cells$, eight $2$-cells, etc.  This isn't a formal proof, but I suspect that a proof wouldn't be hard for anyone willing to spend a few minutes on it.