Homology with Abelian Group Coefficients: What does (-1) mean?

Question

For homology with abelian group $G$ coefficients, the boundary map $\partial$ will have "alternating sum" something like $$\partial_n(\sigma)=\sum_i (-1)^i d_i\sigma$$.

What does -1 mean in this context? Is 1 referring to the identity element of $G$, i.e. $1_G$?

Or is it in the context of considering $G$ as a $\mathbb{Z}$-module, so that $1\cdot g=g$ and $(-1)\cdot g=-g$? (i.e. 1 is not $1_G$ but rather really the integer 1.)

Thanks.

Answer 1

Since $G$ is abelian we can naturally endow it with a structure of a $\mathbb{Z}$ module.

The reason this $(-1)^i$ exists is for reasons of orientation and more concretely to have that $\partial \partial =0$.