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model structure for weak complicial sets

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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 ω-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 ω-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

Δ→OStrωCat→resolveTrimbleωCat,\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 ω-category.

The induced nerve and realization adjunction

TrimbleωCat→N←∣−∣sSetTrimble \omega Cat \stackrel{\overset{|-|}{\leftarrow}}{\underset{N}{\to}} sSet

might maybe even be used to define the transferred model structure on Trimble ω-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 ∣N(C)∣→C serves as a good (cofibrant) replacement in TrimbleωCat for reasonable choices of StrωCat→resolveTrimbleωCat.

References

Section 6 of

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