If $X$ is a pointed space and $A$ is an abelian group, then we can form the space $A[X]$ whose points are finite formal sums $\sum a\_i x\_i$ with $a\_i \in A, x\_i \in X$ subject to some natural relations involving $0,+,$ and the basepoint of $X$. We topologize this space as a quotient of $\bigsqcup A^n \times X^n$ where $A$ has discrete topology. Evidently this is true: $$\pi\_n(A[X],0) \cong \tilde{H}\_n(X;A)$$ if $X$ has the homotopy type of a CW-complex. Can anyone point me to a proof of this fact?



Homotopy groups of linearization of a space
This result appears in M. C. McCord, [*Classifying Spaces and Infinite Symmetric Products*](http://www.ams.org/journals/tran/1969-146-00/S0002-9947-1969-0251719-4/S0002-9947-1969-0251719-4.pdf) published in Transactions of the American Mathematical Society, Vol. 146, (Dec., 1969), pp. 273-298. 


Also, this can be generalized to other generalized homology theories: given any spectrum $E$ one can build out of it a *partial* monoid, $A$, for which a variant of the construction you describe produces, given a space $X$, a new space whose homotopy groups are the reduced $E$-homology of $X$. This is proved in Jacob Mostovoy's paper [*Partial monoids and Dold-Thom functors*](http://arxiv.org/abs/0712.3444).


