nLab localization of a 2-category

Idea

The generalization of the concept of localization of categories from categories to 2-categories.

Definition

Definition

(localization of a 2-category)

Let $\mathcal{C}$ be a 2-category and let $W \subset 1Mor(\mathcal{C})$ be a sub-class of its 1-morphisms.

Then the the 2-localization x is, if it exists,

1. a 2-category $\mathcal{C}[W^{-1}]$;

2. a 2-functor $\mathcal{C} \overset{\gamma}{\longrightarrow} \mathcal{C}[W^{-1}]$

such that

1. $\gamma$ sends all elements of $W$ to an equivalence in the 2-category $\mathcal{C}[W^{-1}]$;

2. $\gamma$ is universal with this property: precomposition with $\gamma$ induces for every 2-category $\mathcal{D}$ an equivalence of 2-categories

   $$(-) \circ \gamma \;\colon\; 2Func(\mathcal{C}[W^{-1}], \mathcal{D}) \overset{\simeq}{\longrightarrow} 2Func(\mathcal{C},\mathcal{D})_{W} \subset 2Func(\mathcal{C}, \mathcal{D})$$

   from the 2-functor 2-category from $\mathcal{C}[W^{-1}]$ to $\mathcal{D}$ to the full sub-2-category of the 2-functor 2-category from $\mathcal{C}$ to $\mathcal{D}$ on those 2-functors that send element of $W$ to equivalences.

Examples

Homotopy 2-category of combinatorial model categories

Definition

(the 2-category of combinatorial model categories)

Write

1. $ModCat$ for the 2-category whose objects are model categories, 1-morphisms are left adjoint functors of Quillen adjunctions and 2-morphisms are natural transformations.

2. $CombModCat \subset ModCat$ for the full sub-2-category on the combinatorial model categories.

Remark

(local presentation of combinatorial model categories)

By Dugger's theorem, we may choose for every $\mathcal{C} \in CombModCat$ a simplicial set $\mathcal{S}$ and a Quillen equivalence

$$\mathcal{C}^p \;\coloneqq\; [\mathcal{S}^{op}, sSet]_{proj,loc} \overset{\simeq_{Qu}}{\longrightarrow} \mathcal{C}$$

from the local projective model structure on sSet-enriched presheaves over $\mathcal{S}$.

Theorem

(the homotopy 2-category of combinatorial model categories)

The 2-localization (Def. )

$$CombModCat\big[QuillenEquivalences^{-1}\big]$$

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 $CombModCat$ and for any $\mathcal{C}, \mathcal{D} \in CombModCat$ its hom-category is the localization of categories

$$CombModCat\big[QuillenEquivalences^{-1}\big](\mathcal{C}, \mathcal{D}) \;\simeq\; ModCat( \mathcal{C}^p, \mathcal{D}^p )\big[\{QuillenHomotopies\}^{-1}\big]$$

of the category of left Quillen functors and natural transformations between local presentations $\mathcal{C}^p$ and $\mathcal{D}^p$ (Remark ) at those natural transformation that on cofibrant objects have components that are weak equivalences (“Quillen homotopies”).

This is the statement of Renaudin 06, theorem 2.3.2.

For $\mathcal{C}$ a 2-category write

1. $\mathcal{C}_0$ for the 1-category obtained by discarding all 2-morphisms;

2. $\pi_0^{iso}(\mathcal{C})$ for the 1-category obtained by identifying isomorphic 2-morphisms.

Proposition

The composite 1-functor

$$CombModCat_0 \longrightarrow \pi_0^{iso}(CombModCat) \overset{\pi_0^{iso}(\gamma)}{\longrightarrow} \pi_0^{iso}( CombModCat[QuillenEquivalences^{-1}] )$$

induced from the 2-localization of Theorem exhibits the ordinary localization of a category of the 1-category $CombModCat$ at the Quillen equivalences, hence Ho(CombModCat).

This is the statement of Renaudin 06, cor. 2.3.8.

Related concepts

• bicategory of fractions

• homotopy 2-category

• localization of an enriched category

References

• Olivier Renaudin, Section 1.2 of: Plongement de certaines théories homotopiques de Quillen dans les dérivateurs, J. Pure Appl. Alg. 213:10 (2009) 1916–1935

  (arXiv:math/0603339, doi:10.1016/j.jpaa.2009.02.014)

Via bicategories of fractions:

• Dorette A. Pronk, Section 2 of: Etendues and stacks as bicategories of fractions, Comp. Math. 102 3 (1996) pp.243-303. (numdam:CM_1996__102_3_243_0)

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]