TITLE: Which $\infty$-groupoids correspond to simplicial abelian groups?
QUESTION [7 upvotes]: 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.

REPLY [4 votes]: 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!