# nLab model structure for homotopy n-types

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

# Contents

## Idea

A model structure for homotopy $n$-types with $n \in \mathbb{N}$ is a model category which presents an (n+1,1)-category of homotopy n-types hence of n-truncated objects in some ambient (∞,1)-category.

For plain homotopy $n$-types (in ∞Grpd/Top) this is a presentation of the collection of n-groupoids. With respect to an arbitrary (∞,1)-topos it is a presentation of n-stacks.

Typically these model structures can be obtained as the left Bousfield localization of model structure that present the ambient (∞,1)-category of all homotopy types.

## References

• J. Cabello, A. Garzon, Quillen’s theory for algebraic models of $n$-types, Extracta mathematica Vol. 9, Num. 1, 42-47 (1994)

• J. Cabello, A. Garzon, Closed model structures for algebraic models of $n$-types, Journal of pure and applied algebra 103 (1995) 287-302

• J. Cabello, Estructuras de modelos de Quillen para categorías que modelan algebraicamente tipos de homotopía de espacios, Ph.D. thesis, Universidad de Granada. (1993)

• Georg Biedermann, On the homotopy theory of n-types (2006) (arXiv:math/0604514)

The case of homotopy 1-types in an (∞,1)-topos, hence of (2,1)-sheaves/stacks is discussed in

Last revised on January 6, 2016 at 08:02:20. See the history of this page for a list of all contributions to it.