nLab model structure for homotopy n-types

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) link

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

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

• 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

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