Homotopy coherent space maps induces homotopy coherent chain complex morphisms

Question

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.

Is it true that $C_*$ is a simplicial functor from $\textrm{Top}$ to $\textrm{Ch}(\mathbb{Z})$?

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:

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?

Answer 1

Be careful with this "simplicial structure on chain complexes". It's not really well-defined, as discussed in the comments below my answer here. 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)