I've come across the following definition. Given some piecewise continuously differentiable curves $\gamma_i, 1 \leq i \leq n$, we define the formal sum
$\gamma = a_1\gamma_1 + ... + a_n\gamma_n$, with each $a_i$ an integer. This is called a cycle.
It says this does not mean addition of the curves. What exactly does it mean? Is it just a sequence of curves?
Given some different kinds of fruit $X_i$, define the formal sum
$$Y=a_1X_1+\dots a_nX_n$$
with each $a_i$ and integer.
So for example let $X_1=$ apple and $X_2=$ banana, $a_1=$ 2 and $a_2=$ 1. Then the corresponding formal sum is
$$Y=2X_1+X_2$$
which could represent two apples and one banana.
That's really all the formal sum is meant to do, to represent a certain collection of curves, some of them occurring perhaps multiple times. You can then add and subtract different collections $Y_1$ and $Y_2$ in the obvious way.
The point of this is that you will eventually define integration over $Y$, which will amount to the sum of the integrals of the curves $\gamma_i$, with multiplicity. In other words, you will define
$$\int_Yf(z)dz=a_1\int_{\gamma_1}f(z)dz+\dots+a_n\int_{\gamma_n}f(z)dz$$
To be clear, it does not represent a sequence of curves or pointwise addition of curves.