Dold-Kan correspondence in the category of symmetric spectra

Question

The Dold-Kan correspondence between the category of simplicial abelian groups and the category of non-negatively graded chain complexes of abelian groups is a classical result. It states that the normalization functor $N$, which sends a simplicial abelian group to a normalized chain complex, possesses an adjoint functor $D$ such that $$N : s(Ab) \to Ch_{\geq 0}(Ab): D$$ is an equivalence of categories.

Also, we know that there is another construction, called the Moore complex functor, $M : s(Ab) \to Ch_{\geq 0}(Ab)$, given by $M(A_*)_n = A_n$ and the differentials are just alternating sums of face maps.

I think there should be an analogue of Dold-Kan correspondence in the category of Symmetric spectra in the sense of Hovey-Shipley-Smith.

Is there any possibility of a Quillen adjoint functor of $M?$

Answer 1

Usually when people talk about Dold-Kan in the context of spectra, they are referring to the "stable Dold-Kan correspondence." The classic Dold-Kan correspondence also works for rings $R$ other than $\mathbb{Z}$, where it says simplicial $R$-modules are equivalent (and Quillen equivalent) to non-negatively graded chain complexes of $R$-modules. The spectra version says modules over the Eilenberg-MacLane ring spectrum $HR$ are Quillen equivalent to unbounded chain complexes of $R$-modules. In this paper, Schwede and Shipley prove this result as Theorem 5.1.6 (even more generally, for $R$ a ringoid, but it works for a ring), via a zig-zag of Quillen equivalences. I assume that when Denis says "it's only true at the level of $\infty$-categories" he means that's the case where you don't have the zig-zag. Indeed, on the model category level, we don't know how to do it without a zig-zag. Shipley later lifted the result to categories of $HR$-algebra spectra and differential graded algebras (i.e. monoids in $Ch(R)$). Even later, Richter and Shipley lifted the result to commutative $HR$-algebras and commutative differential graded algebras. Then Donald Yau and I lifted it to $O$-algebras where $O$ is a colored operad (acting in $HR$-modules and in $Ch(R)$), and in doing so we also recovered the earlier results with a shorter zig-zag.

These results are true for many different choices of model of spectra. Notably, they are true for symmetric spectra, and most of the papers are written in that model.

The OP also mentioned the Moore functor $M$. That is also a Quillen equivalence (see Remark 3.7 here) in the algebraic setting of the question. I view the normalized chains functor $N$ as an improvement of $M$, but you could try to work with $M$ throughout. There are good details here about the extra structure needed on chain complexes so that $M$ is part of an equivalence of categories. You could then mimic the work here to check if this determines a weak monoidal Quillen pair. As far as I'm aware, that would be novel research.

Answer 2

The OP has clarified that their question is as follows:

Is there a (zig-zag) of Quillen equivalence(s) between chain complexes in symmetric spectra, $\mathbf{Ch}(\mathbf{Sp}^{\Sigma})$, and simplicial objects in symmetric spectra, $\mathbf{sSp}^{\Sigma}$?

One issue is that I’m not sure what a chain complex of symmetric spectra even is. The weakest possible interpretation is just a chain complex in the additive category $h\mathsf{Sp}$. Since $h\mathsf{Sp}$ is idempotent complete, the classical Dold-Kan correspondence gives a sort of answer to your question:

There is a canonical equivalence of categories between $\mathbf{Ch}(h\mathsf{Sp})$ and $\mathbf{Fun}(\Delta^{op}, h\mathsf{Sp})$.

However, this is not the greatest result. A simplicial object in the homology category of spectra is pretty far from a simplicial object in spectra. For example, one cannot form a `geometric realization’ of such a thing, so this equivalence has limited use. One would really like a description of the $\infty$-category $\mathsf{Fun}(\Delta^{op}, \mathsf{Sp})$ (which is modeled by the usual model structure on simplicial objects in symmetric spectra, for example).

Now let me return to the original concern: what exactly do we mean by a chain complex of spectra? You wanted to include, as an example, the result of taking a simplicial spectrum and taking the “alternating sum of face maps”. But that doesn’t really make sense: the category of symmetric spectra (or any model of spectra I know of) isn’t literally additive, it only becomes so upon passage to the homotopy category (or, in the infinity categorical sense, after passage to the associated infinity category). After making some choices, the best you could do is supply functorial nullhomotopies of each of the composites, and compatibilities between these, ad infinitum. (One could phrase this as witnessing the vanishing of a bunch of higher Toda brackets, for example).

It has been known for a while that there is a more efficient way to encode the data of a sequence of maps of spectra where ‘all higher Toda brackets vanish’- as a filtered spectrum.

Let’s see how this works at the bottom. Suppose we have a putative chain complex that just consists of a choice of nullhomotopy for the composite $C_2 \to C_1 \to C_0$. This is equivalent to the datum of a map $\Sigma C_2 \to \mathrm{cofib}(C_1 \to C_0)$. Let’s define $F_0:=C_0$, $F_1:= \mathrm{cofib}(C_1 \to C_0)$, and $F_2:= \mathrm{cofib}(\Sigma C_2 \to F_1)$. Then this yields a sequence of maps $F_0 \to F_1 \to F_2$. (If you’re working at the model categorical level, you can arrange for these to be cofibrations). Thinking about this further yields an equivalence of homotopy theories between 2-step diagrams with a chosen nullhomotopy and 2-step filtered spectra. (To go the other way, take cofibers).

Jazzing this process up yields the following special case of the “infinity-categorical Dold-Kan theorem” (Higher Algebra 1.2.4.1):

There is a canonical equivalence of $\infty$-categories $\mathsf{Fun}(\mathbb{Z}_{\ge 0}, \mathsf{Sp}) \simeq \mathsf{Fun}(\Delta^{op}, \mathsf{Sp})$.

Informally, this equivalence assigns to each simplicial spectrum $X_{\bullet}$ the filtered object $\{\mathrm{sk}_nX\}_{n}$ given by its skeletal filtration.

Associated to any filtered spectrum $\{F_kY\}$ one can form an associated graded with $\mathrm{gr}_kY:=\mathrm{cofib}(F_{k-1}Y \to F_{k}Y)$. The defining cofiber sequences then produce maps $\mathrm{gr}_0Y \leftarrow \Sigma^{-1}\mathrm{gr}_1Y\leftarrow \Sigma^{-2}\mathrm{gr}_2Y\leftarrow \cdots$

The composite of any two of these maps is null so this produces, at least, a chain complex in the additive category $h\mathsf{Sp}$. In this way, a filtered spectrum (and hence a simplicial spectrum) is approximated by a ‘chain complex’ of spectra. You can think of a filtered spectrum as encoding a “coherent chain complex”.

I don’t think you can get rid of these coherences. For example, from a simplicial spectrum one can extract the geometric realization. If you could extract the same data from some more naive version of chain complex of spectra, then what would its homotopy groups be? It seems to me one would take homotopy groups of the terms in your “chain complex” and be confronted with a chain complex of graded abelian groups. From here, at best, you could take the homology of that chain complex. In general, that’s very far from the homotopy groups of the geometric realization of the thing we started with.

In the case of modules over an Eilenberg-MacLane spectrum, we still have this mismatch in general (filtered chain complexes lead to spectral sequences too!), but if you start with a simplicial object of ordinary modules then the extra ‘coherences’ needed in the associated ‘chain complex’ can be added in for free because of all the vanishing homotopy groups in sight.