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 into $\Delta^1$. It has eight 2-cells, which we see by looking at the set of subobjects of $\Delta^2$ 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.

Edit: It appears that this is wrong! See the comments below.

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 into $\Delta^1$. It has eight 2-cells, which we see by looking at the set of subobjects of $\Delta^2$ etc.

The nerve of $G_2$ has two $0$-cells, four $1$-cells, eight $2$-cells, etc.

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.

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 into $\Delta^1$. It has eight 2-cells, which we see by looking at the set of subobjects of $\Delta^2$ 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.