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 -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 -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
The induced nerve and realization adjunction
might maybe even be used to define the transferred model structure on Trimble -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 serves as a good (cofibrant) replacement in for reasonable choices of .
Section 6 of