nLab
model structure on simplicial abelian groups

Contents

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Contents

Definition

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

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 QusSet_{Qu}:

(1)sAbGrp proj Quundrlng[]sSet Qu 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.

References

Due to:

Further discussion:

Last revised on July 14, 2021 at 06:50:43. See the history of this page for a list of all contributions to it.