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localization of a 2-category

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Idea

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

Definition

Definition

(localization of a 2-category)

Let ๐’ž\mathcal{C} be a 2-category and let WโŠ‚1Mor(๐’ž)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 ๐’ž[W โˆ’1]\mathcal{C}[W^{-1}];

  2. a 2-functor ๐’žโŸถฮณ๐’ž[W โˆ’1]\mathcal{C} \overset{\gamma}{\longrightarrow} \mathcal{C}[W^{-1}]

such that

  1. ฮณ\gamma sends all elements of WW to an equivalence in the 2-category ๐’ž[W โˆ’1]\mathcal{C}[W^{-1}];

  2. ฮณ\gamma is universal with this property: precomposition with ฮณ\gamma induces for every 2-category ๐’Ÿ\mathcal{D} an equivalence of 2-categories

    (โˆ’)โˆ˜ฮณ:2Func(๐’ž[W โˆ’1],๐’Ÿ)โŸถโ‰ƒ2Func(๐’ž,๐’Ÿ) WโŠ‚2Func(๐’ž,๐’Ÿ) (-) \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 ๐’ž[W โˆ’1]\mathcal{C}[W^{-1}] to ๐’Ÿ\mathcal{D} to the full sub-2-category of the 2-functor 2-category from ๐’ž\mathcal{C} to ๐’Ÿ\mathcal{D} on those 2-functors that send element of WW to equivalences.

Examples

Homotopy 2-category of combinatorial model categories

Definition

(the 2-category of combinatorial model categories)

Write

  1. ModCatModCat 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. CombModCatโŠ‚ModCatCombModCat \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 ๐’žโˆˆCombModCat\mathcal{C} \in CombModCat a simplicial set ๐’ฎ\mathcal{S} and a Quillen equivalence

๐’ž pโ‰”[๐’ฎ op,sSet] proj,locโŸถโ‰ƒ Qu๐’ž \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 ๐’ฎ\mathcal{S}.

Theorem

(the homotopy 2-category of combinatorial model categories)

The 2-localization (Def. )

CombModCat[QuillenEquivalences โˆ’1] 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 CombModCatCombModCat and for any ๐’ž,๐’ŸโˆˆCombModCat\mathcal{C}, \mathcal{D} \in CombModCat its hom-category is the localization of categories

CombModCat[QuillenEquivalences โˆ’1](๐’ž,๐’Ÿ)โ‰ƒModCat(๐’ž p,๐’Ÿ p)[{QuillenHomotopies} โˆ’1] 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 ๐’ž p\mathcal{C}^p and ๐’Ÿ p\mathcal{D}^p (Remark ) at those natural transformation that on cofibrant objects have components that are weak equivalences (โ€œQuillen homotopiesโ€).

This is the statement of Renaudin 06, theorem 2.3.2.

For ๐’ž\mathcal{C} a 2-category write

  1. ๐’ž 0\mathcal{C}_0 for the 1-category obtained by discarding all 2-morphisms;

  2. ฯ€ 0 iso(๐’ž)\pi_0^{iso}(\mathcal{C}) for the 1-category obtained by identifying isomorphic 2-morphisms.

Proposition

The composite 1-functor

CombModCat 0โŸถฯ€ 0 iso(CombModCat)โŸถฯ€ 0 iso(ฮณ)ฯ€ 0 iso(CombModCat[QuillenEquivalences โˆ’1]) 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 CombModCatCombModCat at the Quillen equivalences, hence Ho(CombModCat).

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

References

Last revised on July 6, 2018 at 16:57:57. See the history of this page for a list of all contributions to it.