nLab
canonical model structure on 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

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 GrpdGrpd – a functor f:CDf : C \to D – is

Proposition

Equipped with this structure Grpd natGrpd_{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 natGrpd_{nat} is the restriction of the canonical model structure on Cat from categories to groupoids.

See natural model structure for more.

Definition

Let

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

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

Proposition

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

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

and Grpd natGrpd_{nat} is the transferred model structure obtained from sSet QuillensSet_{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)K(n)-local duality for finite groups and groupoids , Topology 39, (2000).

The model structure on functors with values in Grpd natGrpd_{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)