Which $\infty$-groupoids correspond to simplicial abelian groups?

Question

Kan complexes model $\infty$-groupoids, so since every simplicial abelian group is a Kan complex, every simplicial abelian group yields an $\infty$-groupoid. What sort of $\infty$-groupoids do you get?

(A natural guess would be grouplike symmetric monoidal $\infty$-groupoids, but I think that's not the correct answer roughly because $\mathbb{Z}$ is not the sphere spectrum.)

I realize that this question can be taken to be tautological, but I'd like an answer that I could say specialize $2$-groupoids described by the bicategory axioms.

I'd also like to understand what you get if you translate the functor taking simplicial set to the corresponding simplicial abelian group across the homotopy hypothesis.

Answer 1

Basically, you're asking which (weak) homotopy types can be modelled by simplicial sets which carry the structure of a simplicial abelian group.

Every simplicial abelian group splits as a product of Eilenberg MacLane spaces. $$X \simeq \prod_{n \in \mathbb N} K(\pi_i X, n).$$ This is a consequence of the Dold-Kan correspondence, and the fact that every chain complex of abelian groups is quasi-isomorphic to its homology. Conversely, you can use Dold-Kan to realize every product of Eilenberg-MacLane spaces as a simplicial abelian group.

Answer 2

$\newcommand{\Z}{\mathbf{Z}} \newcommand{\Mod}{\mathrm{Mod}}$Note that if $\Mod^{\geq 0}_\Z$ denotes the category of connective $\mathrm{H}\Z$-module spectra, then by the Dold-Kan correspondence and the Schwede-Shipley theorem, there are equivalences of categories $$\Mod^{\geq 0}_\Z \simeq \mathrm{Ch}_{\geq 0}(\Z) \simeq \mathrm{Fun}(\Delta^{op},\mathrm{Ab}) = s\mathrm{Ab}.$$ (You could interpret "category" as either $\infty$-categories or model categories, but the latter means that "equivalences" has to be replaced with "zig-zag of Quillen equivalences". I'll just sweep this distinction under the rug.) Since simplicial abelian groups are Kan complexes, we're asking for a characterization of the image of $\Mod^{\geq 0}_\Z$ under the equivalence of categories between connective spectra and infinite loop spaces (which are grouplike $\mathbf{E}_\infty$-objects in spaces).

Here is one such characterization. Let's model grouplike infinite loop spaces $X$ as functors $X:\mathrm{Fin}_\ast\to \mathrm{Top}$ such that $\pi_0 \mathrm{Map_{Top}}(Y,X)$ is an abelian group for all spaces $Y$ (i.e., $X$ is grouplike) and such that the map $X([n])\to X([1])^n$ is an equivalence. Such an object should be in the image of $\Mod^{\geq 0}_\Z$ iff it is somehow "strictly commutative". One way to characterize this is as follows. Let $\Lambda$ denote the full subcategory of the category of abelian groups spanned by the groups $\Z^n$ with $n\geq 0$, so there is a functor $\mathrm{Fin}_\ast\to \Lambda$. Then an infinite loop space is in the image of $\Mod^{\geq 0}_\Z$ if and only if the functor $\mathrm{Fin}_\ast\to \mathrm{Top}$ classifying it factors through a finite-product-preserving functor $\Lambda \to \mathrm{Top}$. In other words, $\Mod^{\geq 0}_\Z$ is equivalent to the full subcategory spanned by the grouplike objects in the category $\mathrm{Fun^{prod}}(\Lambda, \mathrm{Top})$. This is a very strong condition to impose on an infinite loop space; for example, $\mathbf{C}P^\infty$ admits such a factorization, but $BU$ (with either the additive or multiplicative infinite loop space structure) doesn't.

Answer 3

The paper HHA.5(1), 2003, pp.49–52 gives a few categories equivalent to chain complexes, including cubical abelian groups with connections, and cubical $\omega$-groupoids with connections. There are also the simplicial $T$-complexes of N. Ashley, referenced [Ash88] in the bibliography of NAT-book. A simplicial $T$-complex is a simplicial complex with in each dimension a spacial kind of simplex called thin satisfying Dakin's axioms: 1) degenerate elements are thin; 2) every horn has a unique thin filler; 3) if every face but one of a thin simplex is thin then so also is the remaining face. (cf Note 222 p. 476 of the NAT book).

These structures of $T$-complex do not model all homotopy types, but only the "linear" ones (no quadratic information, such as Whitehead products) but, like linear algebra, it has its uses!