nLab
model structure on simplicial groupoids

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

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

Contents

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.)

Definition

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.

Properties

Proposition

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)

Remark

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.

References

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

  • Paul Goerss and Frederick Jardine?, 1999, Simplicial Homotopy Theory, number 174 in Progress in Mathematics, Birkhauser. (ps)
Revised on October 18, 2014 22:37:23 by Urs Schreiber (185.26.182.26)