model category

for ∞-groupoids

# Contents

## Idea

The notion of weak complicial set is a model for (the omega-nerve of) the notion of weak ω-category. The model structure for weak $\omega$-categories is a model category structure on the category of stratified simplicial sets, such that cofibrant-fibrant objects are precisely the weak complicial sets. This model structure may therefore be understood as a presentation of the (∞,1)-category of weak $\omega$-categories.

Urs Schreiber: It would be interesting to see which algebraic definitions of higher categories coul be equipped with a model category structure induced from this model structure for weak complicial sets under the corresponding natural nerve and realization adjunction, maybe even as a transferred model structure.

For instance I imagine that there is a good cosimplicial Trimble ω-category

$\Delta \stackrel{O}{\to }\mathrm{Str}\omega \mathrm{Cat}\stackrel{\mathrm{resolve}}{\to }\mathrm{Trimble}\omega \mathrm{Cat}\phantom{\rule{thinmathspace}{0ex}},$\Delta \stackrel{O}{\to} Str \omega Cat \stackrel{resolve}{\to} Trimble \omega Cat \,,

where the first step is Street’s orientals and the second step is an embedding that regards a strict ω-category as a suitably resolved equivalent Trimble $\omega$-category.

The induced nerve and realization adjunction

$\mathrm{Trimble}\omega \mathrm{Cat}\stackrel{\stackrel{\mid -\mid }{←}}{\underset{N}{\to }}\mathrm{sSet}$Trimble \omega Cat \stackrel{\overset{|-|}{\leftarrow}}{\underset{N}{\to}} sSet

might maybe even be used to define the transferred model structure on Trimble $\omega$-catgeories (but I haven’t checked at all if the relevant conditions have a chance of being satisfied, though). In any case I would expect that the counit of the adjunction $\mid N\left(C\right)\mid \to C$ serves as a good (cofibrant) replacement in $\mathrm{Trimble}\omega \mathrm{Cat}$ for reasonable choices of $\mathrm{Str}\omega \mathrm{Cat}\stackrel{\mathrm{resolve}}{\to }\mathrm{Trimble}\omega \mathrm{Cat}$.

## References

Section 6 of

Revised on April 6, 2010 14:32:36 by Urs Schreiber (131.211.235.55)