#
nLab

model (∞,1)-category

### Context

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

**(∞,1)-category theory**

## Background

## Basic concepts

## Universal constructions

## Local presentation

## Theorems

## Models

#### Model category theory

**model category**, model $\infty$-category

## 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

#### specific

### 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

The notion of *model $(\infty,1)$-category* (or *model $\infty$-category*, for short) is the $(\infty,1)$-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