The ordinary homology, hence the standard $n$-th homology group of a CW-complex$X$ is isomorphic to the $n$-th homotopy group of the free topological commutative monoid on $X$, which is an infinite symmetric product: that is, the colimit$\mathrm{Sym}^\infty X$ (denoted also $SP^\infty$) of the symmetric powers $\mathrm{Sym}^N X = X*X*...*X = (X\times X\times ...\times X)/\Sigma_N$ of $X$:

The Mayer-Vietoris sequence for homology is a consequence of applying $\pi_*(-)$ to the homotopy pullback square resulting from the application of $\mathrm{Sym}^\infty$ to the homotopy pushout square formed by the inclusions of the intersection, $A \cap B$, of two subspaces $A$ and $B$ of a space $X$ into $A$ and $B$.

References

The original article is

A. Dold, R. Thom, Quasifaserungen und unendliche symmetrische Produkte , Ann. Math. (2) 69 (1959), 239–281. (pdf)

Last revised on October 19, 2014 at 10:29:09.
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