Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
The generalization of the concept of localization of categories from categories to 2-categories.
(localization of a 2-category)
Let be a 2-category and let be a sub-class of its 1-morphisms.
Then the the 2-localization x is, if it exists,
a 2-category ;
such that
sends all elements of to an equivalence in the 2-category ;
is universal with this property: precomposition with induces for every 2-category an equivalence of 2-categories
from the 2-functor 2-category from to to the full sub-2-category of the 2-functor 2-category from to on those 2-functors that send element of to equivalences.
(the 2-category of combinatorial model categories)
Write
for the 2-category whose objects are model categories, 1-morphisms are left adjoint functors of Quillen adjunctions and 2-morphisms are natural transformations.
for the full sub-2-category on the combinatorial model categories.
(local presentation of combinatorial model categories)
By Dugger's theorem, we may choose for every a simplicial set and a Quillen equivalence
from the local projective model structure on sSet-enriched presheaves over .
(the homotopy 2-category of combinatorial model categories)
The 2-localization (Def. )
of the 2-category of combinatorial model categories (Def. ) at the Quillen equivalences exists. Up to equivalence of 2-categories, it has the same objects as and for any its hom-category is the localization of categories
of the category of left Quillen functors and natural transformations between local presentations and (Remark ) at those natural transformation that on cofibrant objects have components that are weak equivalences (โQuillen homotopiesโ).
This is the statement of Renaudin 06, theorem 2.3.2.
For a 2-category write
for the 1-category obtained by discarding all 2-morphisms;
for the 1-category obtained by identifying isomorphic 2-morphisms.
The composite 1-functor
induced from the 2-localization of Theorem exhibits the ordinary localization of a category of the 1-category at the Quillen equivalences, hence Ho(CombModCat).
This is the statement of Renaudin 06, cor. 2.3.8.
Via bicategories of fractions:
Another construction:
Maria E. Descotte, Eduardo J. Dubuc, M. Szyld, A localization of bicategories via homotopies, Theory and Applications of Categories 35 23 (2020) 845-874 [tac:2020 MR4112764]
Maria E. Descotte, Eduardo J. Dubuc, M. Szyld, Model bicategories and their homotopy bicategories, Adv. Math. B 404 (2022) 108455 [arXiv:1805.07749 doi:10.1016/j.aim.2022.108455]
Last revised on June 6, 2024 at 10:34:18. See the history of this page for a list of all contributions to it.