nLab model structure on presheaves of simplicial groupoids

Idea

The model structure on presheaves of (Dwyer-Kan) simplicial groupoids (sSet-enriched groupoids) is one of the models for ∞-stack (∞,1)-toposes. It is a slight variant on the model structure on simplicial presheaves. (At that link more general information is collected).

For various applications it is useful to

• use simplicial groupoids instead of Kan complexes or simplicial sets;

• use a model structure on (pre)sheaves with values in simplicial groupoids instead of the model structure on simplicial presheaves (see there for motivation on why to consider these models in the first place).

An example is the discussion of principal infinity-bundles in section 3 of (Jardine & Luo)

Definition/Properties

Proposition

Write $(G \dashv \bar W) \colon Grpd^\Delta \leftrightarrow sSet$ for the Quillen adjunction discussed at model structure on simplicial groupoids. This directly prolongs to an adjunction on presheaves

$$(G \dashv \bar W) \colon [C^{op}, Grpd^\Delta] \stackrel{\overset{G}{\leftarrow}}{\underset{\bar W}{\to}} [C^{op}, sSet]_{inj} \,.$$

The transferred model structure along $\bar W$ on $[C^{op}, Grpd^\Delta]$ of the global injective model structure on simplicial presheaves exists on $[C^{op}, sSet, Grpd^\Delta]$: fibrations and weak equivalences are those that become global injective fibrations and weak equivalences, respectively, under $\bar W$.

This appears as (LBK, theorem 3.10).

Related concepts

• model structure on simplicial sets

  • model structure on simplicial presheaves

• model structure on simplicial groupoids

References

A model structure on sheaves with values in simplicial groupoids is due to

• Andre Joyal and Myles Tierney, Strong stacks and classifying spaces, in: Category theory

  (Como, 1990), volume 1488 of Lecture Notes in Math., pages 213–236. Springer, Berlin (1991) (web)

• Andre Joyal and Myles Tierney, On the homotopy theory of sheaves of simplicial groupoids, Math. Proc. Cambridge Philos. Soc. 120 2 (1996) 263-290

A model structure on presheaves with values in simplicial groupoids:

• Zhi-Ming Luo, Peter Bubenik, Peter Kim, Closed model categories for presheaves of simplicial groupoids and presheaves of 2-groupoids (arXiv)

A model structure on simplicial sheaves of groupoids is discussed in

• Sjoerd Crans, Quillen closed model structure for sheaves, J. Pure Appl. Algebra 101 (1995), 35-57 (web)

• John F. Jardine, Luo, Higher order principal bundles (web)

• John F. Jardine, §9.4 in: Local homotopy theory, Springer Monographs in Mathematics (2015) [doi:10.1007/978-1-4939-2300-7]