Schreiber
model structure on omega categories internal to spaces

Idea

The point is to introduce ωCategories(Spaces)Sheaves(CartesianSpaces,ωCategories)\omega Categories(Spaces) \simeq Sheaves(CartesianSpaces, \omega Categories) with weak equivalences being the ordinary weak equivalences of ω\omega-functors directly internalized into Spaces=SheavesSpaces = Sheaves (which yields stalkwise weak equivalences in ωCategories(Sets)\omega Categories(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 ωGroupoids\omega 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 ωGroupoids(Spaces)\omega Groupoids(Spaces) by setting

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

  • fibrations are the globally kk-surjective ω\omega-functors;

  • the path object of CC is C I=hom(I,C)C^I = hom(I,C) with II 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 ω\omega-groupoids to ω\omega-categories.

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

Revised on December 16, 2008 19:40:58 by Urs Schreiber (134.100.222.156)