On the meaning of a linear combination of simplexes I'm struggling a bit to understand the concept of a chain in Geometry/Topology, as a linear combination of simplexes, and even more to understand it geometrically (if it possible). So, let's start simply.
Let $\Delta = [a,\,b,\,c]$ be the triangle generated by the points $a,\,b,\,c$ in our space, such that $\partial \Delta$ is the chain $[a,\,b] + [b,\,c] - [a,\,c]$.
If we interpret the boundary $\partial \Delta$ of this triangle as "paths", with a orientation, it's easy to see that $\partial \Delta$ is the polygonal path connecting $a$ to $c$, with $b$ in between. Ok, but what about the chain $[a,\,b] + 2\,[b,\,c] - 5\,[a,\,c]$, how could we interpret it geometrically?
Now, let's go to $\mathbb{R}^n$; considering continuous paths of the form $c_i \colon [0,\,1] \longrightarrow \mathbb{R}^n$, is there a intuitive way of understanding chains of the form $\Gamma = \sum_i n_i \, c_i$, with $n_i \in \mathbb{Z}$?
In my point of view, the integral coeficients of $\Gamma$ implies "how many times" we travel the path $c_i$, i.e., if $\Gamma = 3\,c_1 - c_2 + 2\,c_3$, we travel $3$ times along $c_1$, then one time along $c_1$ (in the opposite way), and then $2$ times along $c_3$.
My goal is to understand properly the Generalized Stoke's Theorem, as well the homologic version of Cauchy's Theorem, and both requires the language of chains. Thanks in advance!
 A: This is not something you should try to overthink. Nonetheless there's a couple of simple straightforward ways that I like to think of the 1-chain $[a,b] + 2 [b,c] - 5 [a,c]$.
One is almost purely formal. First, attached to the oriented segment $[a,b]$ you observe a ghostly floating coefficient $1$. Also, attached to the oriented segment $[b,c]$ you observe a ghostly floating coefficient $2$. And, attached to the oriented segment $[a,c]$, you observe a ghostly floating coefficient $-5$.
Another, less imaginatively spectral way might be like this. When you look closely at the oriented segment $[b,c]$ you actually observe 2 copies of it. When you look closely at the oriented segment $[a,c]$ you actually observe $-5$ copies of it, which leads to the issue of what a "negative" copy means, but perhaps you can think of it as 5 anti-copies, where an "anti"copy and an ordinary copy annihilate each other leaving nothing, just as $-5$ and $5$ add up to nothing. Finally, when you look closely at the oriented segment $[a,b]$ you don't see anything out of the ordinary at all, just $1$ ordinary copy.
And your point of view, counting the how many times to travel, is just fine too.
Oh, and here's one more: think of each simplex as a pipe, and the coefficient as the amount of fluid per unit time flowing through that pipe. So $[a,b] + 2 [b,c] - 5 [a,c]$ is 1 fluid unit per time unit flowing along $[a,b]$ from $a$ to $b$, 2 fluid units per time unit flowing along $[b,c]$ from $b$ to $c$, and $5$ fluid units per time unit flowing along $[a,c]$ from $c$ to $a$. What I like about this point of view is that the boundary of the chain makes physical sense:
$$\partial \left( [a,b] + 2 [b,c] - 5 [a,c] \right) = 4a -b -3c
$$
means that at $a$ you see 4 fluid units gushing outward per unit time, and at $b$ you see 1 fluid unit sucked inward per unit time; and at $c$ you see 3 fluid units sucked inward per unit time.
