Formal finite sum for integration on k-chains This question is related to the integration of differential forms on chains, as exposed in the document at: http://www.math.upenn.edu/~ryblair/Math%20600/papers/Lec17.pdf . The author gives the following definitions:
Let U be an open subset of $\mathbb{R}^n$. A singular k-cube in U is a continuous map
$c:[0,1]^k\rightarrow U$
A (singular) k-chain in U is a formal finite sum of singular k-cubes in U with integer coefficients, such as $2c^1+ 3c^2−4c^3$.
Such a decomposition is used later for the integration of a differential form $\omega$, in:
$\int_c \omega = \sum a_i\int_{c_i} \omega$
What is, in simple terms, the definition (or at least an intuition) of a formal finite sum of functions in this context ? Is a Free Abelian Group involved, with which group operation ?

Singular $k$-chains form a free abelian group generated by singular $k$-cubes. You can define integral of differential forms over singular $k$-cubes, and the integral over chains is just the $\mathbb{Z}$-linear extension of the integral maps over cubes.
