What do the elements of the chains of a simplicial complex represent? I've just started to learn homology and I don't quite understand why we define chains the way we do. For a simplicial complex $S$ we define $C_k$ to be the $k$-chains on $S$ given by an abelian group of integer linear combinations of $k$-simplices. Are these linear combinations formal and chosen because they give us a nice group structure to work with or do they actually correspond to something geometrically?
 A: Short answer: Yes, the linear combinations are formal because they give you nice group structure.
Longer answer: Historically, cohomology came before homology. (See comments below.) In the modern pedagogy, we work with homology then dualize chain complexes and introduce cohomology, and as if by magic, this dual functor has all the same qualities and has extra structure like the cup product and other operations.
But, one important thing is lost in this presentation. It's intuitive to add, scale, and take linear combinations of scalar-valued functions $\Delta \to \mathbb{Z}$. The original advent of cohomology had to do with studying the properties of integration of differential forms around closed curves that enclosed singularities, where the scalar ring was $\mathbb{R}$. Think about contour integration or residues in complex analysis. This approach yields de Rham cohomology.
But if $f: \Delta \to \mathbb{Z}$ is a ring homomorphism (i.e. $\mathbb{Z}$-linear), then
$$
\sum_i n_i f(\sigma_i) = f \biggl( \sum_i n_i \sigma_i \biggr),  
$$
where the second expression involves taking the linear combination on the chains, i.e. considering elements of the dual to $\mathbb{Z}[\Delta]$. There's a bit of subtlety about finiteness that's addressed in this question, but in spirit, this explains why formal linear combinations of chains and the homology theory are worthwhile to study.
