The forgetful functor from topological abelian monoids to topological spaces has a left adjoint, the infinite symmetric product $\text{SP}$. The Dold-Thom theorem asserts that if $X$ is a CW-complex, then $H_n(X) \cong \pi_n(\text{SP}(X))$; in other words, the singular homology of $X$ is precisely the homotopy of a "linearized" version of $X$.
If you want to phrase this theorem in a more combinatorial setting, you can replace $X$ with a simplicial set, hence replace $\text{SP}(X)$ with the free simplicial abelian group on $X$.
Theorem (Dold-Kan): The category of simplicial abelian groups is equivalent to the category of chain complexes concentrated in non-negative degree.
This theorem explains, in a precise sense, why the machinery of homological algebra can describe properties of topological spaces. For me it really explains what a chain complex "is": it's a linearization of a combinatorialization of a topological space.
