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The framework of Quillen's model categories gives us a very general way of defining things as derived functors. For instance, in this way one can realise the singular homology as Andre-Quillen homology so it is also a derived functor.

My question is the opposite. For a derived functor in this sense, could we realise as the (co)homology of spaces. In particular, given a ring $R$ and modules $M,N$ so that $M\otimes_RN$ makes sense, is there a space $T(R,M,N)$ so that $$H_i(T(R,M,N);\mathbb{Z})\cong\mathrm{Tor}_i^R(M,N)$$ is true?

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    $\begingroup$ Left derived functors like Tor tend to be the homotopy groups of simplicial objects. In particular, $\mathrm{Tor}_i^R(M,N)=\pi_i(B(M,R,N))$. These homotopy groups get transformed into homology groups of chain complexes by the Dold-Kan correspondence. $\endgroup$
    – Connor Malin
    Commented Apr 15, 2023 at 15:22
  • $\begingroup$ @ConnorMalin Is $B(M,R,N)$ the simplicial bar construction? $\endgroup$
    – Li Guanyu
    Commented Apr 15, 2023 at 15:33
  • $\begingroup$ Since you start with an abelian category ($R$-modules) you can take any flat resolution $X$ of $M$, take the simplicial flat $R$-module $KX$ under the Dold-Kan functor $K$, then the simplcial $R$-module $KX\otimes_RN$ and then the homotopy of the underlying simplicial set. It is the converse, "one can realise the singular homology as Andre-Quillen homology", the one that I do not understand. Could you provide some details or reference? $\endgroup$
    – A.G
    Commented Apr 20, 2023 at 5:57

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