nLab canonical model structure on groupoids

Idea

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.

Definition

Definition

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

• a weak equivalence iff it is an equivalence of categories hence a weak homotopy equivalence of groupoids,

• a fibration iff it is an isofibration,

• a cofibration iff it is an isocofibration, hence injective on objects.

Proposition

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

• combinatorial

  (in particular cofibrantly generated),

• proper,

• simplicial

  (induced under the simplicial nerve by the canonical enrichment over itself),

• cartesian monoidal.

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 $Grpd_{can}$ is combinatorial. Finally the cartesian model structure follows just as for the canonical model structure on Cat, see there.

Properties

Relation to canonical model structure on $Cat$

Remark

The model structure $Grpd_{nat}$ 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 $sSet$

Consider the pair of adjoint functors

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

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

One readily checks that:

Proposition

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

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

(cf. Hollander 2001, Cor. 2.3)

In fact:

Proposition

$Grpd_{can}$ is the transferred model structure obtained from $sSet_{Qu}$ under (1).

(cf. Hollander 2001, Lem. 2.4)

Related concepts

• canonical model structure

• canonical model structure on Cat

• canonical model structure on $Grpd$

  • model structure for n-groupoids

  • model structure for ∞-groupoids

• canonical model structure on Operad

References

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 15 1 (1970) 103-132 [doi:10.1016/0021-8693(70)90089-X, pdf]

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

• Donald W. Anderson, p. 783 (12 of 37) in: Fibrations and geometric realization, Bull. Amer. Math. Soc. 84 5 (1978) 765-788 [euclid:1183541139]

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

• Aldridge Bousfield, Homotopy Spectral Sequences and Obstructions , Israel Journal of Math., 66 1-3 (1989) 54-105 [doi:10.1007/BF02765886

Proofs are spelled out in:

• André Joyal, Myles Tierney, Thm. 2 (p. 221) Strong stacks and classifying spaces, in: Category Theory Lecture Notes in Mathematics 1488, Springer (1991) 213-236 [doi:10.1007/BFb0084222]

  (in the generality of internal groupoids in a topos, hence of stacks)

• Neil Strickland, §6.1 of: $K(n)$-local duality for finite groups and groupoids , Topology 39 4 (2000) [arXiv:math/0011109, doi:10.1016/S0040-9383(99)00031-2]

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

• Sharon Hollander, §2.1 in: A homotopy theory for stacks, Israel Journal of Mathematics 163 1 (2008) 93-124 [arXiv:math.AT/0110247, doi:10.1007/s11856-008-0006-5]

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

• Amit Sharma, Picard groupoids and $\Gamma$-categories, Cahiers LXIV 3 (2023) [arXiv:2002.05811, pdf]