nLab canonical model structure on groupoids

Contents

Idea

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

Let Grpd be the 1-category of small groupoids and functors between them. Say thata morphism in $Grpd$ — a functor $f \colon C \longrightarrow D$ — is

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

cofibrantly generated;
left proper
a simplicial model category with respect to the natural sSet-enriched category structure induced by the canonical enrichment over itself, under the nerve.

This is originally due to (Anderson 78) and (Bousfield 89). A detailed discussion is in (Strickland 00, section 6). In the context of the model structure for (2,1)-sheaves it appears as (Hollander 01, theorem 2.1).

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

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.

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.

Some aspects (like the pullback stability of fibrations of groupoids in its prop. 2.8) appeared in

Ronnie Brown, Fibrations of groupoids, Journal of Algebra
Volume 15, Issue 1, May 1970, Pages 103-132

The full model category structure appears originally in

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

and

Aldridge 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

Last revised on May 1, 2023 at 09:37:20. See the history of this page for a list of all contributions to it.