nLab model structure on simplicial abelian groups

Definition

The category of simplicial abelian groups carries the structure of a model category $sAbGrp_{proj}$ whose weak equivalences and fibrations are those of the underlying morphisms in the classical model structure on simplicial sets, hence whose

• weak equivalences are the simplicial weak homotopy equivalences,

• fibrations are the Kan fibrations

of underlying simplicial sets.

Hence the free/underlying-adjoint functors (where the free functor produces free simplicial abelian groups $\mathbb{Z}[-]$) is a Quillen adjunction with the classical model structure on simplicial sets $sSet_{Qu}$:

(1)$$sAbGrp_{proj} \underoverset {\underset{undrlng}{\longrightarrow}} {\overset{\mathbb{Z}[-]}{\longleftarrow}} {\bot_{\mathrm{Qu}}} sSet_{Qu}$$

Properties

Monoidal structure

With respect to the degreewise tensor product of abelian groups this is a monoidal model category (Schwede & Shipley 2003, p. 312 (26 of 48)).

Dold-Kan correspondence

The Dold-Kan correspondence yields (see there for more) a Quillen equivalence to the projective model structure on connective chain complexes.

(Quillen 67, Section II.4 item 5, see also Schwede-Shipley 03, section 4.1, p.17, Jardine 03, Lemma 1.5).

In fact, much more is true: all five classes of maps in a model category (weak equivalences, (acyclic) cofibrations, and (acyclic) fibrations) are preserved and reflected by both of these equivalences. That is to say, each model structure is obtained from the other one by transferring it along the corresponding equivalence of categories.

Related concepts

• Dold-Kan correspondence

• model structure on cosimplicial abelian groups

• model structure on simplicial algebras

References

Due to:

• Daniel Quillen, Section II.4 in: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer 1967(doi:10.1007/BFb0097438)

Further discussion:

• Stefan Schwede, Brooke Shipley, Equivalences of monoidal model categories , Algebr. Geom. Topol. 3 (2003), 287–334 (arXiv:math.AT/0209342, euclid:euclid.agt/1513882376)

• J. F. Jardine, Lemma 1.5 in: Presheaves of chain complexes, K-theory 30.4 (2003): 365-420 (pdf, pdf)