nLab model structure for homotopy n-types



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

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

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

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

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

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




A model structure for homotopy nn-types with nn \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 nn-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.


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

  • J. Cabello, A. Garzon, Closed model structures for algebraic models of nn-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

Last revised on June 27, 2023 at 19:37:30. See the history of this page for a list of all contributions to it.