Follow up question about homology with coefficients

Question

In my previous question, I asked how one could say that the elements of the $p$-th singular chain group with coefficients in an abelian group, $\Delta_p(X;G) := \Delta_p \otimes_{\mathbb Z} G$, $G \in \mathsf {Ab}$, can be written as formal sums $$\sum_\sigma a_\sigma \sigma,$$ where $a_\sigma \in G$. In the comments, with the help of peter a g, I arrived at the conclusion that $$\Delta_p(X;G) \simeq \bigoplus_{i\in I}G,$$ where $I$ is the set of all singular $p$-simplices on $X \in \mathsf {Top}$. My question is now, how do we identify the direct sum above with the set of formal $G$-linear combinations of $p$-simplices?

Answer 1

You've said that you've arrived at the conclusion that $\Delta_p(X;G) \cong \bigoplus_{i \in I} G$. That is, to each singular simplex $\sigma$, we assign a copy of $G$. An element of this direct sum is given by choosing an element of $G$ for each $i$, such that only finitely many of the elements chosen are non-zero. In other words, each element gives us a function $I \to G$, where only finitely many of the image of $I$ are non-zero. We can call this function $g:I \to G$ and denote $g(i) = g_i$.

Now, for each element in $\Delta_p$, you can write the corresponding formal sum by $\sum g(i) i = \sum g_i i$. The formal sum really is just associating each simplex with its weight in $G$.

As I mentioned in my comments above, this is no different than looking at $(a,b,c) \mapsto ae_1 + be_2 + ce_3$.

I think you may be looking too hard into the "multiplication" $g \cdot \sigma$. It really is just a weight / a "formal" multiplication. A free $G$-module by definition is just the $G$-module generated by a set, so the module multiplication is just putting them next to each other.

In the $\mathbb{Z}$ case, $3 \sigma$ doesn't mean much either. You can think of it as 3 copies of the simplex, but in the same way you can think of $g$ copies fo the simplex in a formal sense.