on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
The model structure on simplicial groups is a presentation of the (∞,1)-category of ∞-groups in ∞Grpd $\simeq$ Top. See group object in an (∞,1)-category.
There is a model category structure on the category $sGrp$ of simplicial groups where a morphism is
is a weak equivalence if the underlying morphism is a weak equivalence in the standard model structure on simplicial sets;
is a fibration if the underlying morphism is a Kan fibration of simplicial sets;
is a cofibration if it has the left lifting property with respect to all trivial fibrations.
(Quillen 67, II 3.7, see also Goerss-Jardine 99, V)
Forming simplicial loop space objects and classifying spaces gives a Quillen equivalence
(Goerss-Jardine 99, Chapter V, Prop. 6.3)
The model structure on simplicial groups is due to
Further discussion is in
Last revised on October 26, 2020 at 05:29:35. See the history of this page for a list of all contributions to it.