#
nLab

model structure on presheaves over a test category

Contents
### Context

#### Model category theory

**model 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

For $C$ a test category, the canonical structure of a category with weak equivalences on the category of presheaves over $C$ lifts to the structure of a model category. All of these are models for the standard homotopy theory (the homotopy category of ∞Grpd).

## Examples

## References

The model structure is due to

Further developments are in

- Rick Jardine,
*Categorical homotopy theory*, Homot. Homol. Appl. **8** (1), 2006, pp.71–144, (HHA, pdf).

Last revised on October 15, 2020 at 13:00:13.
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