nLab canonical model structure on 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 canonical model 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.



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


Equipped with this structure Grpd natGrpd_{nat} is a model category which is

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

See natural model structure for more.



(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.


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.


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.


  • 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 StricklandK(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

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