The point is to introduce with weak equivalences being the ordinary weak equivalences of -functors directly internalized into (which yields stalkwise weak equivalences in ).
The task is to characterize the corresponding homotopy category in terms of $\omega$-anafunctors.
But there is probably a better way, using the fact that according to Lafont et. al’s article on the model structure on omega-categories this category has the property of having ana-inverses.
The main theorem of strategy A was supposed to be:
Theorem We obtain the structure of a category of fibrant objects on by setting
weak equivalence to be the essentially -surjective -functors for all (recall this means that certain local sections exist);
fibrations are the globally -surjective -functors;
the path object of is with the interval groupoid.
We need to
give a more detailed discussion of the precise details of the “stalkwise” versus “on covers” conditions for the local properties;
check the main theorem above;
see to which degree this extends from -groupoids to -categories.
Bigger task: try to check of the Brown-Golasinski model structure on crossed complexes is compatible with the folk model structure on -categories under the equivalence of crossed complexes with -groupoids.