nLab
model structure on strict omega-categories

Idea

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

It resticts to the model structure on strict ω-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 StrωGrpd along the forgetful functor

$$U:\mathrm{Str}\omega \mathrm{Grpd}\to \mathrm{Str}\omega \mathrm{Cat}$$U : Str \omega Grpd \to Str \omega Cat

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

This is proven in (AraMetayer).

References

The model structure on strict ω-groupoids was introduced in

• Ronnie Brown, Marek Golasinski; A model structure for the homotopy theory of crossed complexes (numdam)

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

• Yves Lafont, Francois Métayer, Krzysztof Worytkiewicz, A folk model structure on omega-cat (arXiv) Adv. Math. 224 (2010), no. 3, 1183–1231.

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

• Francois Métayer, Cofibrant complexes are free (arXiv)

• Francois Métayer, Resolutions by polygraphs (tac)

The relation betwee then 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 $\mathrm{\infty }$-groupoids revisited (arXiv) Homology Homotopy Appl. 13 (2011), no. 1, 121–142.