Schreiber
model structure on omega categories internal to spaces

Idea

The point is to introduce $\omega \mathrm{Categories}(\mathrm{Spaces})\simeq \mathrm{Sheaves}(\mathrm{CartesianSpaces},\omega \mathrm{Categories})$ with weak equivalences being the ordinary weak equivalences of $\omega$-functors directly internalized into $\mathrm{Spaces}=\mathrm{Sheaves}$ (which yields stalkwise weak equivalences in $\omega \mathrm{Categories}(\mathrm{Sets})$).

The task is to characterize the corresponding homotopy category in terms of $\omega$-anafunctors.

Strategy A was to make use of the fact that $\omega \mathrm{Groupoids}$ satisfy the axioms of a category of fibrant objects which allows to make use of Kenneth Brown's theorm.

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 $\omega \mathrm{Groupoids}(\mathrm{Spaces})$ by setting

• weak equivalence to be the essentially $k$-surjective $\omega$-functors for all $k$ (recall this means that certain local sections exist);

• fibrations are the globally $k$-surjective $\omega$-functors;

• the path object of $C$ is $C^I=\mathrm{hom}(I,C)$ with $I$ the interval groupoid.

To do list

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 $\omega$-groupoids to $\omega$-categories.

• Bigger task: try to check of the Brown-Golasinski model structure on crossed complexes is compatible with the folk model structure on $\omega$-categories under the equivalence of crossed complexes with $\omega$-groupoids.