model category, model -category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of -categories
Model structures
for -groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant -groupoids
for rational -groupoids
for rational equivariant -groupoids
for -groupoids
for -groups
for -algebras
general -algebras
specific -algebras
for stable/spectrum objects
for -categories
for stable -categories
for -operads
for -categories
for -sheaves / -stacks
The term simplicial groupoid is often used for a simplicial object in the category Grpd of groupoids whose simplicial set of objects is simplicially constant. We will write for the category of such simplicial groupoids.
(BEWARE: perhaps a more accurate term for this concept is simplicially enriched groupoid, and conceptually it is often the enriched category structure that is useful. Because of this it is advisable to check the use being made of the term when consulting the literature. This is more fully discussed at simplicial category.)
There is a model category structure on whose
fibrations are the morphisms such that
for every object of and every morphism in there is a morphism of such that ;
for every object in the induced morphism is a Kan fibration.
weak equivalences are morphisms such that
induces in isomorphism on connected components ;
for each object of the induced morphism is a weak equivalence in the model structure on simplicial groups or equivalently in the model structure on simplicial sets.
This appears for instance as (GoerssJardine, theorem 7.8)
When restricted to simplicial groupoids of the form for a simplicial group and its delooping groupoid this produces a standard presentation of looping and delooping for infinity-groups. See there for more details.
The model structure is discussed after corollary 7.3 in chapter V of
Last revised on October 18, 2014 at 22:37:23. See the history of this page for a list of all contributions to it.