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 Grpd of groupoids (with functors between them) is a presentation of the (2,1)-category of groupoids, functors and natural isomorphisms.

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



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


Equipped with the classes from Def. Grpd is a model category Grpd canGrpd_{can}, which is

The claim that the classes in Def. indeed give a model structure is due to Anderson 1978, repeated by Bousfield 1989, both without proof. Proofs appear in Joyal & Tierney 1991, Thm. 2 (there in the generality of groupoids internal to topoi, hence: of stacks) and Strickland 2000. Properness follows immediately (by this Prop.) because all objects are evidently bifibrant. In the context of the model structure for (2,1)-sheaves the structure is considered, Hollander 2001, theorem 2.1 who states also the cofibrant generation and simplicial enrichment, without proof. Since Grpd is a locally presentable category (for instance by this Prop, observing that groupoids are the models of a limit sketch, namely given by diagrams as shown at internal category) this implies that Grpd canGrpd_{can} is combinatorial. Finally the cartesian model structure follows just as for the canonical model structure on Cat, see there.


Relation to canonical model structure on CatCat


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

See at canonical model structure for more.

Relation to classical model structure on sSetsSet

Consider the pair of adjoint functors

(1)(τN):GrpdNτsSet (\tau \dashv N) \,\colon\, Grpd \underoverset { \underset{N}{\longrightarrow} } { \overset{\tau}{\longleftarrow} } { \bot } sSet

where NN is the simplicial nerve with values in the category sSet of simplicial sets.

One readily checks that:


With the canonical model structure on GrpdGrpd (from Prop. ) and the classical model structure on simplicial sets, (?)NerveAdjunction is a Quillen adjunction

(τN):Grpd can QuNτsSet Qu. (\tau \dashv N) \,\colon\, Grpd_{can} \underoverset { \underset{N}{\longrightarrow} } { \overset{\tau}{\longleftarrow} } { \bot_{\mathrlap{Qu}} } sSet_{Qu} \,.

(cf. Hollander 2001, Cor. 2.3)

In fact:


Grpd canGrpd_{can} is the transferred model structure obtained from sSet Qu sSet_{Qu} under (1).

(cf. Hollander 2001, Lem. 2.4)


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

The existence of the model structure is stated (without proof) in:

and (by referencing Anderson, still without proof) in:

Proofs are spelled out in:

The model structure on functors with values in Grpd natGrpd_{nat} (a model structure for (2,1)-sheaves):

A model structure on CatCat but localized such as to make the fibrant objects be groupoids:

Last revised on November 2, 2023 at 07:42:11. See the history of this page for a list of all contributions to it.