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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 $X$.