model category

for ∞-groupoids

# Contents

## Idea

The natural model category structure on the 1-category of groupoids and functors is a presentation of the (2,1)-category of groupoids, functors and natural isomorphisms.

This is one flavor of the various natural model structures on categories and higher categories.

## Definition

###### Definition

Let Grpd be the 1-category of small groupoids and functors between them. Say a morphism in $Grpd$ – a functor $f : C \to D$ – is

###### Proposition

Equipped with this structure $Grpd_{nat}$ is a model category which is

This is originally due to (Anderson) and (Bousfield). A detailed discussion is in section 6 of (Strickland). In the context of the model structure for (2,1)-sheaves it appears as (Hollander, theorem 2.1).

## Properties

###### Observation

The model structure $Grpd_{nat}$ is the restriction of the canonical model structure on Cat from categories to groupoids.

See natural model structure for more.

###### Definition

Let

$(\tau \dashv N) : Grpd \stackrel{\overset{\tau}{\leftarrow}}{\underset{N}{\to}} sSet$

be the pair of adjoint functors, where $N$ is the nerve of groupoids with values in sSet.

###### Proposition

With the natural model structure on $Grpd$ and the standard model structure on simplicial sets this is a Quillen adjunction

$(\tau \dashv N) : Grpd_{nat} \stackrel{\overset{\tau}{\leftarrow}}{\underset{N}{\to}} sSet_{Quillen} \,.$

and $Grpd_{nat}$ is the transferred model structure obtained from $sSet_{Quillen}$ under this adjunction.

## References

This model category structure appears originally in

• D.W. Anderson, Fibrations and Geometric Realizations , Bull. Am. Math Soc. 84, 765-786, (1978), 765-786.

and

• A. K. Bousfield, Homotopy Spectral Sequences and Obstructions , Israel Journal of Math., Vol.66, Nos.1-3, (1989), 54-105.

A detailed description is in section 6 of

• Neil Strickland $K(n)$-local duality for finite groups and groupoids , Topology 39, (2000).

The model structure on functors with values in $Grpd_{nat}$ (a model structure for (2,1)-sheaves) is discussed in

Revised on November 23, 2012 19:28:00 by Urs Schreiber (131.174.41.241)