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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?

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  • $\begingroup$ I suppose the empty simplex is the geometric realisation of the empty simplicial set $\endgroup$
    – FShrike
    Commented Dec 14, 2022 at 18:31
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    $\begingroup$ There are times when you want to consider the augmented chain complex, and there are times when you don't. I don't think there is a mathematical argument for which is "more natural." (When you get to the study of cohomology, for example, an important feature is that it has a ring structure, and the identity element lives in degree 0. If you insist on always using the augmented chain complex, then the degree 0 cohomology of a path connected space will be zero, so you will end up dealing with rings without unit, and that is inconvenient.) $\endgroup$ Commented Dec 15, 2022 at 0:02
  • $\begingroup$ Very similar to math.stackexchange.com/q/4355056 $\endgroup$
    – Paul Frost
    Commented Dec 15, 2022 at 8:04

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