model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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
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
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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.
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
Last revised on June 27, 2023 at 19:37:30. See the history of this page for a list of all contributions to it.