# nLab model structure on presheaves of simplicial groupoids

model category

for ∞-groupoids

## Theorems

#### $\left(\infty ,1\right)$-Category theory

(∞,1)-category theory

# Contents

## Idea

The model structure on presheaves of simplicial 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

An example is the discussion of principal infinity-bundles in section 3 of (JardineLuo)

## Definition/Properties

###### Proposition

Write $\left(G⊣\overline{W}\right):{\mathrm{Grpd}}^{\Delta }↔\mathrm{sSet}$ for the Quillen adjunction discussed at model structure on simplicial groupoids. This directly prolongs to an adjunction on presheaves

$\left(G⊣\overline{W}\right):\left[{C}^{\mathrm{op}},{\mathrm{Grpd}}^{\Delta }\right]\stackrel{\stackrel{G}{←}}{\underset{\overline{W}}{\to }}\left[{C}^{\mathrm{op}},\mathrm{sSet}{\right]}_{\mathrm{inj}}\phantom{\rule{thinmathspace}{0ex}}.$(G \dashv \bar W) : [C^{op}, Grpd^\Delta] \stackrel{\overset{G}{\leftarrow}}{\underset{\bar W}{\to}} [C^{op}, sSet]_{inj} \,.

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

This appears as (LBK, theorem 3.10).

## 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):263–290, 1996.

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

• 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)

• Rick Jardine, Luo, Higher order principal bundles (web)

Revised on August 9, 2011 10:19:57 by Urs Schreiber (89.204.137.76)