Although Hatcher calls the empty set a simplex in his book Algebraic topology on page 110 he seems to assume that $0$-simplices $[v_0]$ have no faces at all in the definition of $\Delta$-complexes on page 102. Therefore, the condition that the restriction $\sigma_\alpha$ to a face of $\Delta^n$ is one of the maps $\sigma_\beta$ for some $\beta$ is vacuously fulfilled.
Wouldn't it be much more natural (or even logically necessary) to consider the empty simplex as the only face of a $0$-simplex because $[\hat v_0]=[\;]$ is the empty simplex?
Then one of the $\sigma_\beta$ is the empty map $\emptyset\to X$ and $\Delta_{-1}(X)$ is the free group with a single generator $\mathbb Z$. The chain complex in the definition of simplicial homology is then automatically augmented.
The follow-up question about the faces of the empty simplex is more debatable but for homology it is probably best to say that it has no faces which makes $\Delta_{-2}(X)=\{0\}$.
Is there anything speaking against this?
