# nLab model structure on strict omega-categories

Contents

### Context

#### Model category theory

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 $\infty$-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

# Contents

## Idea

The model structure on strict $\omega$-categories is a model category structure that presentes the (∞,1)-category of strict ∞-categories.

It resticts to the model structure on strict ∞-groupoids.

These structures also go by the name canonical model structure or folk model structure.

## Properties

###### Observation

Every object is fibrant. The acyclic fibrations are precisely the functors that are k-surjective functors for all $k \in \mathbb{N}$.

###### Theorem

The transferred model structure on Str∞Grpd? along the forgetful functor

$U : Str \omega Grpd \to Str \omega Cat$

exists and coincides with the model structure on strict ∞-groupoids defined in (BrownGolasinski).

This is proven in (AraMetayer).

## References

The model structure on strict ∞-groupoids was introduced in

The model structure on strict $\omega$-categories was discussed in

Dicussion of cofibrant resolution in this model structure by polygraphs/computad is in

The relation between the model structure on strict $\omega$-categories and that on strict $\omega$-groupoids is established in

• Dimitri Ara, Francois Metayer, The Brown-Golasinski model structure on strict $\infty$-groupoids revisited (arXiv) Homology Homotopy Appl. 13 (2011), no. 1, 121–142.

Last revised on November 10, 2014 at 20:37:04. See the history of this page for a list of all contributions to it.