nLab model structure on simplicial groups



Model category theory

model category, model \infty -category



Universal constructions


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 \infty-algebras

specific \infty-algebras

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

Group Theory



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 sGrpsGrp of simplicial groups where a morphism is

(Quillen 67, II 3.7, see also Goerss-Jardine 99, V)


Further model category properties



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 FF of the boundary and horn-inclusions into simplicial simplices:

I={F(i n):F(Δ[n])F(Δ[n])} n I \;=\; \big\{ F(i_n) \,\colon\, F(\partial \Delta[n]) \hookrightarrow F(\Delta[n]) \big\}_{n \in \mathbb{N}}
J={F(j n k):F(Λ k[n])F(Δ[n])} n +,0kn J \;=\; \big\{ F(j_n^k) \,\colon\, F(\Lambda_k[n]) \hookrightarrow F(\Delta[n]) \big\}_{ {n \in \mathbb{N}_+,} \atop {0 \leq k \leq n} }


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 sGrpsSet QusGrp \to sSet_{Qu} with left adjoint the degreewise free group-functor F:sSetsGrpF \,\colon\,sSet \to sGrp. Now since sSet QusSet_{Qu} is cofibrantly generated with generating (acyclic) cofibrations i ni_n (j n kj_n^k) (this Prop.) it follows (by this Prop.) that so is sGrp, with generating (acyclic) cofibrations their images under FF, 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

f n:𝒢 n𝒢 nF(S n) f_n \,\colon\, \mathcal{G}_n \xhookrightarrow{\phantom{---}} \mathcal{G}_n \star F(S_n)

in a compatible way.

(Goerss & Jardine (1999), p. 270; Speirs (2015), §A.4)


The class of almost free maps (Prop. ) is closed under

  1. isomorphism

  2. pushout

  3. transfinite composition

Speirs (2015), Prop. A.4.1


In the model structure on simplicial groups (Prop. ):

  1. Every almost free homomorphism (Def. ) is a cofibration.

  2. 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)} nI = \{F(i_n)\}_{n \in \mathbb{N}}, which means that the cofibrations of sGrpsGrp are the retracts of the relative cell complexes (i.e. of the transfinite compositions of pushouts) of the F(i n)F(i_n). Therefore it is now sufficient to see that

Cell({F(i n)})AlmostFreeMapsandCell({F(i n)})AlmostFreeMaps. Cell\big( \{ F(i_n) \} \big) \;\; \supset \;\; AlmostFreeMaps \, \;\;\;\;\;\;\;\; \text{and} \;\;\;\;\;\;\;\; Cell\big( \{ F(i_n) \} \big) \;\; \subset \;\; AlmostFreeMaps \,.

The first statement is Goerss & Jardine (1999), V Prop. 1.9. The second statement follows by Prop. after observing that F(i n)F(i_n) itself is almost free.


All cofibrations in sGrp are monomorphisms, hence the underlying maps are cofibrations in sSet Qu 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 QusSet_{Qu}) are closed under retracts.

Relation to reduced simplicial sets


(Quillen equivalence between simplicial groups and reduced simplicial sets)

Forming simplicial loop space objects and simplicial classifying spaces gives a Quillen equivalence

(ΩW¯):sGrpsSet 0 \big( \Omega \dashv \overline{W} \big) \;\colon\; sGrp \stackrel{\overset{}{\longleftarrow}}{\longrightarrow} sSet_0

with the model structure on reduced simplicial sets.

(Goerss-Jardine 99, Chapter V, Prop. 6.3)

Relation to simplicial groupoids


Forming simplicial delooping groupoids constitutes a fully faithful functor of 1-categories from simplicial groups to sSet-enriched groupoids (DK-simplicial groupoids):

sGrp sSet-Grpd 𝒢 B𝒢. \array{ sGrp &\hookrightarrow& sSet\text{-}Grpd \\ \mathcal{G} &\mapsto& \mathbf{B}\mathcal{G} \mathrlap{\,.} }

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.


The model structure on simplicial groups is due to:

Further detailed discussion:


  • Martin Speirs, Model Categories with a view towards rational homotopy theory, MSc thesis, Copenhagen (2015) [pdf, pdf]

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.