#
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 $(\infty,1)$-categories

## Model structures

### for $\infty$-groupoids

for ∞-groupoids

### for equivariant $\infty$-groupoids

### for rational $\infty$-groupoids

### for rational equivariant $\infty$-groupoids

### for $n$-groupoids

### for $\infty$-groups

### for $\infty$-algebras

#### general

#### specific

### for stable/spectrum objects

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

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

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

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

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

# Contents

## 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

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.

## 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.