nLab
model structure on simplicial groupoids

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 $s\mathrm{Grpd}$ 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.)

Definition

There is a model category structure on $\mathrm{sGrpd}$ whose

• fibrations are the morphisms $f:H\to K$ such that

  1. for every object $x$ of $H$ and every morphism $\omega :f(x)\to y$ in $K_0$ there is a morphism $\hat{\omega }:x\to z$ of $H_0$ such that $f(\hat{\omega })=\omega$;

  2. for every object $x$ in $H$ the induced morphism $f:H(x,x)\to K(f(x),f(x))$ is a Kan fibration.

• weak equivalences are morphisms $f:H\to K$ such that

  1. $f$ induces in isomorphism on connected components ${\pi }_{0}f:{\pi }_{0}H\to {\pi }_{0}K$;

  2. for each object $x$ of $H$ the induced morphism $H(x,x)\to K(f(x),f(x))$ is a weak equivalence in the model structure on simplicial groups or equivalently in the model structure on simplicial sets.

Properties

Proposition

We have a Quillen adjunction

$$(G⊣\overline{W}):{\mathrm{Grpd}}^{\Delta }\stackrel{\stackrel{G}{\leftarrow }}{\underset{\overline{W}}{\to }}{\mathrm{sSet}}_{\mathrm{Quillen}}$$(G \dashv \bar W) : Grpd^\Delta \stackrel{\overset{G}{\leftarrow}}{\underset{\bar W}{\to}} sSet_{Quillen}

where both $G$ and $\overline{W}$ preserve all weak equivalences.

This appears for instance as (GoerssJardine, theorem 7.8)

Remark

When restricted to simplicial groupoids of the form $(BG{)}_{•}$ for $G_{•}$ a simplicial group and $B{G}_n$ its delooping groupoid this produces a standard presentation of looping and delooping for infinity-groups. See there for more details.

Related concepts

• model structure on simplicial sets

• model structure on presheaves of simplicial groupoids

References

The model structure is discussed after corollary 7.3 in chapter V of

• Paul Goerss and J. F. Jardine, 1999, Simplicial Homotopy Theory, number 174 in Progress in Mathematics, Birkhauser. (ps)