model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
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
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 “projective” model category structure on the category $sGrp$ of simplicial groups where a morphism is
a weak equivalence if the underlying morphism is a weak equivalence in the classical model structure on simplicial sets, hence a simplicial weak homotopy equivalence;
a fibration if the underlying morphism is a fibration in the classical model structure on simplicial sets, hence a Kan fibration;
a cofibration if it has the left lifting property with respect to all acyclic fibrations.
(Quillen 67, II 3.7, see also Goerss-Jardine 99, V)
Every object in the projective model structure (Prop. ) is fibrant.
This statement amounts to saying that the underlying simplicial set of any simplicial group is a Kan complex, which is the case by Moore’s theorem (here).
The model structure on sGrp from Prop. is cofibrantly generated, with generating (acyclic) cofibrations the images under the degreewise free group-functor $F$ of the boundary and horn-inclusions into simplicial simplices:
By definition, the model structure on simplicial groups is the right transferred model structure from the classical model structure on simplicial sets, along the forgetful functor $sGrp \to sSet_{Qu}$ with left adjoint the degreewise free group-functor $F \,\colon\,sSet \to sGrp$. Now since $sSet_{Qu}$ is cofibrantly generated with generating (acyclic) cofibrations $i_n$ ($j_n^k$) (this Prop.) it follows (by this Prop.) that so is sGrp, with generating (acyclic) cofibrations their images under $F$, as claimed.
(almost free maps)
A homomorphims of simplicial groups is called almost free if it is degreewise the inclusion into the amalgamation with a free group
in a compatible way.
The class of almost free maps (Prop. ) is closed under
In the model structure on simplicial groups (Prop. ):
Every cofibration is a retract of an almost free map
The first statement is proven in Goerss & Jardine (1999) Cor. 1.10. For the second statement, use with Prop. , that the model category is cofibrantly generated by $I = \{F(i_n)\}_{n \in \mathbb{N}}$, which means that the cofibrations of $sGrp$ are the retracts of the relative cell complexes (i.e. of the transfinite compositions of pushouts) of the $F(i_n)$. Therefore it is now sufficient to see that
The first statement is Goerss & Jardine (1999), V Prop. 1.9. The second statement follows by Prop. after observing that $F(i_n)$ itself is almost free.
All cofibrations in sGrp are monomorphisms, hence the underlying maps are cofibrations in $sSet_{Qu}$.
Since the almost free maps (Def. ) are manifestly monomorphisms and since all cofibrations are retracts of almost free maps (Prop. ) this follows form the fact that injections (cofibrations in $sSet_{Qu}$) are closed under retracts.
(Quillen equivalence between simplicial groups and reduced simplicial sets)
Forming simplicial loop space objects and simplicial classifying spaces gives a Quillen equivalence
(Goerss-Jardine 99, Chapter V, Prop. 6.3)
Forming simplicial delooping groupoids constitutes a fully faithful functor of 1-categories from simplicial groups to sSet-enriched groupoids (DK-simplicial groupoids):
With respect to the above model structure (Prop. ) and the model structure on simplicial groupoids, this functor clearly preserves the weak equivalences and fibrations. However, it does not have a left adjoint and thus fails to be a right Quillen functor.
Borel model structure (presenting infinity-actions of simplicial groups)
The model structure on simplicial groups is due to:
Further detailed discussion:
Review:
See also:
Last revised on July 28, 2023 at 06:27:23. See the history of this page for a list of all contributions to it.