# nLab model structure on presheaves of simplicial groupoids

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of $(\infty,1)$-categories

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

for stable $(\infty,1)$-categories

for $(\infty,1)$-operads

for $(n,r)$-categories

for $(\infty,1)$-sheaves / $\infty$-stacks

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

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

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

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

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).

## 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)

Last revised on August 9, 2011 at 10:19:57. See the history of this page for a list of all contributions to it.