nLab
model (∞,1)-category

Context

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

Model category theory

model category, model \infty -category

Definitions

Morphisms

Universal constructions

Refinements

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

specific

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

Contents

Idea

The notion of model (,1)(\infty,1)-category (or model \infty-category, for short) is the ( , 1 ) (\infty,1) -categorification of that of model category.

Where the classical model structure on simplicial sets is an archetypical example of a model category, so simplicial \infty -groupoids (“simplicial spaces”, bisimplicial sets) form an archetypical example of a model \infty-category. In this example, a fundamental application of the theory says, for instance, that geometric realization preserves homotopy pullbacks of homotopy Kan fibrations (see there).

References