For a commutative ring $R$, there is a "free topological $R$-module" functor $X \mapsto RX$, left adjoint to the forgetful map from topological $R$-modules to topological spaces. (A topological $R$-module is a topological abelian group $A$ together with a continuous map $R \times A \to A$ satisfying the obvious axioms.)

Let $\mathrm{Sin}_p(RX)$ be the set of continuous maps $\sigma: \Delta^p \to RX$. Then $\mathrm{Sin}_\bullet(RX)$ is naturally a simplicial $R$-module, and comes with a homomorphism from the free $R$-module on the simplicial set $\mathrm{Sin}_\bullet(X)$. Hence there is a map of chain complexes $$C_*(X;R) \to N(\mathrm{Sin}_\bullet(RX)),$$ whose domain is the usual singular chains and the codomain is the chain complex corresponding to the simplicial $R$-module we just constructed.

Has the corresponding homology functor been studied? For nice enough spaces it must agree with singular homology, by some variant of the Dold-Thom theorem.


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