It is an elementary fact that a map $f:X \to Y$ between spaces induces a chain complex morphism $f\_\* : C\_\*(X) \to C\_\*(Y) $. This allows one to transfer 1-category theoretic arguments from spaces to associated chain complexes.


When dealing with $\infty$ stuff, sometimes a bit more is needed. Let $\textrm{Sing}(\textrm{Top})$ be the simplicially enriched category of topological spaces with


$$Map\_{\textrm{Top}}(X, Y) \_n = Hom\_{\textrm{Top}}(X \times \Delta^n, Y) $$


And let $\textrm{Ch}(\mathbb{Z}) $ be the simplicially enriched category of chain complexes with


$$Map\_{\textrm{Ch}(\mathbb{Z})}(C, D) = Hom\_{\textrm{Ch}(\mathbb{Z})}(C \otimes \mathbb{Z}[\Delta^n], D) $$


where $\mathbb{Z}[\Delta^n]$ is the simplicially chain complex of $\Delta^n$: it has in degree $k$ the span of its $k$ dimensional faces.



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> Is it true that $C\_\*$ is a simplicial functor from $\textrm{Top}$ to $\textrm{Ch}(\mathbb{Z})$?
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I thought this was almost tautological, but there are some Alexander Whitney and Eilenberg-Zilber maps involved that are not straightforward. A reference would be highly appreciated. I couldn't find this in HTT, but maybe it's me. As a remark, let me say the above functor factors explicitly through simplicial abelian complexes, so the crucial point is probably in the Dold-Kan part.


In case this is not true:



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> Informally, how one can transfer $\infty$ arguments from spaces to chain complexes? For example, is there a quasi-isomorphic functor to $C\_\*$ that becomes simplicial?
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Homotopy coherent space maps induces homotopy coherent chain complex morphisms
Be careful with this "simplicial structure on chain complexes". It's not really well-defined, as discussed in the comments below my answer [here](https://mathoverflow.net/questions/382151/homotopy-coherent-colimits-in-chain-complexes/382154#382154). Also see my remark at the end of this post.


In particular, the answer to your question, as stated, is "not really".


But that's not an issue : the point is that for simplicial model categories, you can either take the homotopy coherent nerves on the subcategory of fibrant-cofibrant objects, *or* take the underlying ordinary model category and invert its weak equivalences (as an $\infty$-category).


The two processes yield canonically equivalent things (1.3.4.20. in *Higher Algebra*).


In particular you don't need a simplicial structure to talk about underlying $\infty$-categories of model categories, nor for induced functors.


Indeed, $C\_\* : Top\to Ch(\mathbb Z)$ sends weak equivalences to quasi-isomorphisms, so it induces a functor from the localized $\infty$-category $Top[weq^{-1}]$ to the localized $\infty$-category $Ch(\mathbb Z)[qis^{-1}]$. These are respectively the $\infty$-category of spaces and the derived $\infty$-category of $\mathbb Z$.


If one wants the stronger statement of having a simplicial functor, one can instead proceed as follows:
notice that $C\_\*$ actually factors as the Dold-Kan correspondance precomposed with $X\mapsto\mathbb Z[Sing(X)]$ as a functor $Top\to sAb$. Now $sAb$ (simplicial abelian groups) *is* a simplicial model category and $sAb\to Ch\_{\geq 0}(\mathbb Z)$ is an equivalence of categories, and now this functor $\mathbb Z[-]$ is also a composite of $Top\to sSet \to sAb$.


The second functor, $sSet\to sAb$ is simplicial by design and $Top\to sSet$ is a Quillen equivalence (in particular an equivalence on underlying $\infty$-categories), but it's also a simplicial functor. So up to replacing $Ch\_{\geq 0}(\mathbb Z)$ by $sAb$, you can get a simplicial functor.


**Remark:** Of course, $Ch\_{\geq 0}(\mathbb Z)\simeq sAb$ so you *can* transport the simplicial model structure on $sAb$ to one on $Ch\_{\geq 0}(\mathbb Z)$. What I mean is that the simplicial structure on $Ch\_{\geq 0}(\mathbb Z)$ looks slightly weird from this perspective, e.g. it is not compatible with the monoidal structure or the internal hom's of $Ch\_{\geq 0}(\mathbb Z)$. In particular I'm not sure you can get away with such a thing over all chain complexes $Ch(\mathbb Z)$. Hence the warning.


Note that this second approach might be needed, or at the very least make things easier if you want to prove some basic properties of the induced $\infty$-functor : e.g. that it is a left adjoint (and in particular preserves all (homotopy, but this is implicit when talking about $\infty$-categories) colimits)


