nLab
canonical model structure on groupoids

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 $\mathrm{Grpd}$ – a functor $f:C\to D$ – is

• a weak equivalence if it is an equivalence of categories;

• a fibrations if it is an isofibration;

• a cofibration if it is an injection on objects.

Proposition

Equipped with this structure ${\mathrm{Grpd}}_{\mathrm{nat}}$ 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) 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 ${\mathrm{Grpd}}_{\mathrm{nat}}$ is the restriction of the canonical model structure on Cat from categories to groupoids.

See natural model structure for more.

Definition

Let

$$(\tau ⊣N):\mathrm{Grpd}\stackrel{\stackrel{\tau }{\leftarrow }}{\underset{N}{\to }}\mathrm{sSet}$$(\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.

Proposition

With the natural model structure on $\mathrm{Grpd}$ and the standard model structure on simplicial sets this is a Quillen adjunction

$$(\tau ⊣N):{\mathrm{Grpd}}_{\mathrm{nat}}\stackrel{\stackrel{\tau }{\leftarrow }}{\underset{N}{\to }}{\mathrm{sSet}}_{\mathrm{Quillen}}.$$(\tau \dashv N) : Grpd_{nat} \stackrel{\overset{\tau}{\leftarrow}}{\underset{N}{\to}} sSet_{Quillen} \,.

and ${\mathrm{Grpd}}_{\mathrm{nat}}$ is the transferred model structure obtained from ${\mathrm{sSet}}_{\mathrm{Quillen}}$ under this adjunction.

Related concepts

• canonical model structure on Cat

• canonical model structure on $\mathrm{Grpd}$

  • model structure for n-groupoids

  • model structure for ∞-groupoids

• canonical model structure on Operad

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 $K(n)$-local duality for finite groups and groupoids , Topology 39, (2000).

The model structure on functors with values in ${\mathrm{Grpd}}_{\mathrm{nat}}$ (a model structure for (2,1)-sheaves) is discussed in

• Sharon Hollander, A homotopy theory for stacks (arXiv:math.AT/0110247)