TITLE: What does 2v mean in the context of Simplicial Homology
QUESTION [9 upvotes]: Consider this picture (taken from Hatcher pg 106). Elements of $\Delta_n(X)$, called $n$-chains, can be written as finite formal sums $\sum_\alpha n_\alpha e^n_\alpha$ with coefficients $n_\alpha\in\mathbb{Z}$, and $e^n_\alpha$ are open $n$-simplices.
I can guess that $2e$ means the edge going around anticlockwise twice (is that correct?)
However, I don't know what $2v$ would represent? Or for that matter $-v$?
Thanks for enlightening!

REPLY [7 votes]: Formal sums are just that: they are form, not content. You shouldn't think of $2[v]$ as anything more than the number $2$ written next to the symbol $[v]$.
Now, as you realized, there's a special way you can think about $2[e]$ for the case that $[e]$ is an edge (which points from one vertex to itself); loosely, as a path that traces this edge twice. This is a map $\pi_1(X, [e]) \to H_1(X)$ called the Hurewicz map, and indeed "trace the edge twice" maps to $2[e]$ in this case, so the geometric way you've thought about it isn't a complete coincidence.