nLab model structure on simplicial groups

Idea

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.

Definition

Proposition

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)

Properties

Further model category properties

Fibrations

Proposition

Every object in the projective model structure (Prop. ) is fibrant.

Proof

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).

Cofibrations

Proposition

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:

$$I \;=\; \big\{ F(i_n) \,\colon\, F(\partial \Delta[n]) \hookrightarrow F(\Delta[n]) \big\}_{n \in \mathbb{N}}$$

$$J \;=\; \big\{ F(j_n^k) \,\colon\, F(\Lambda_k[n]) \hookrightarrow F(\Delta[n]) \big\}_{ {n \in \mathbb{N}_+,} \atop {0 \leq k \leq n} }$$

Proof

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.

Definition

(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

$$f_n \,\colon\, \mathcal{G}_n \xhookrightarrow{\phantom{---}} \mathcal{G}_n \star F(S_n)$$

in a compatible way.

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

Proposition

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

1. isomorphism

2. pushout

3. transfinite composition

Speirs (2015), Prop. A.4.1

Proposition

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

Proof

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

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

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

Corollary

All cofibrations in sGrp are monomorphisms, hence the underlying maps are cofibrations in $sSet_{Qu}$.

Proof

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.

Relation to reduced simplicial sets

Proposition

(Quillen equivalence between simplicial groups and reduced simplicial sets)

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

$$\big( \Omega \dashv \overline{W} \big) \;\colon\; sGrp \stackrel{\overset{}{\longleftarrow}}{\longrightarrow} sSet_0$$

with the model structure on reduced simplicial sets.

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

Relation to simplicial groupoids

Remark

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

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

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.

Related concepts

• model structure on simplicial groupoids

• Borel model structure (presenting infinity-actions of simplicial groups)

References

The model structure on simplicial groups is due to:

• Daniel Quillen, II §3.7 of: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer (1967) [doi:10.1007/BFb0097438]

Further detailed discussion:

• Paul Goerss, J. F. Jardine, chapter V of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (doi:10.1007/978-3-0346-0189-4, webpage)

Review:

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

See also:

• Hans-Joachim Baues, Quadratic Homology, Transactions of the American Mathematical Society 351 2 (1999) [jstor:117809, pdf]