nLab model structure on simplicial groupoids



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


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 sGrpds 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.)


There is a model category structure on sGrpdsGrpd whose

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

    1. for every object xx of HH and every morphism ω:f(x)y\omega : f(x) \to y in K 0K_0 there is a morphism ω^:xz\hat \omega : x \to z of H 0H_0 such that f(ω^)=ωf(\hat \omega) = \omega;

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

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

    1. ff induces in isomorphism on connected components π 0f:π 0Hπ 0K\pi_0 f : \pi_0 H \to \pi_0 K;

    2. for each object xx of HH the induced morphism H(x,x)K(f(x),f(x))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.



We have a Quillen adjunction

(GW¯):Grpd ΔW¯GsSet Quillen (G \dashv \bar W) : Grpd^\Delta \stackrel{\overset{G}{\leftarrow}}{\underset{\bar W}{\to}} sSet_{Quillen}

where both GG and W¯\bar W preserve all weak equivalences.

This appears for instance as (GoerssJardine, theorem 7.8)


When restricted to simplicial groupoids of the form (BG) (B G)_\bullet for G G_\bullet a simplicial group and BG nB G_n 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.