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Ho(CombModCat)

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Idea

The concept of model categories is one way of formulating the concept of certain classes of homotopy theories or (∞,1)-categories. One way to make this precise while staying strictly within the context of 1-category theory is to consider the homotopy category of the (very large) category of model categories of (left) Quillen functors between them, hence its localization of a category at the Quillen equivalences.

This should be particularly well-behaved for the sub-category CombModCatCombModCat of combinatorial model categories. Due to Dugger's theorem, it should be true that

Ho(CombModCat)CombModCat[QuillenEquivs 1]Ho(Pr(,1)Cat) Ho(CombModCat) \;\coloneqq\; CombModCat\big[QuillenEquivs^{-1}\big] \;\simeq\; Ho(Pr(\infty,1)Cat)

is equivalent to the homotopy category of an (infinity,1)-category of Pr(∞,1)Cat, the (∞,1)-category of locally presentable (∞,1)-categories and (∞,1)-colimit-preserving (∞,1)-functors between them. At least when the latter is formalized in terms of derivators, then this is proven in Renaudin 06, see Corollary below.

Details

Definition

(the 2-category of combinatorial model categories)

Write

  1. ModCatModCat for the 2-category whose objects are model categories, whose 1-morphisms are left Quillen functors and 2-morphisms are natural transformations.

  2. CombModCatModCatCombModCat \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 𝒞CombModCat\mathcal{C} \in CombModCat a simplicial set 𝒮\mathcal{S} and a Quillen equivalence

𝒞 p[𝒮 op,sSet] proj,loc Qu𝒞 \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 𝒮\mathcal{S}.

Proposition

(the homotopy 2-category of combinatorial model categories)

The 2-localization of a 2-category

CombModCat[QuillenEquivs 1] CombModCat\big[QuillenEquivs^{-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 𝒞,𝒟CombModCat\mathcal{C}, \mathcal{D} \in CombModCat its hom-category is the localization of categories

CombModCat[QuillenEquivs 1](𝒞,𝒟)ModCat(𝒞 p,𝒟 p)[{QuillenHomotopies} 1] CombModCat\big[QuillenEquivs^{-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 𝒞 p\mathcal{C}^p and 𝒟 p\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.

Proposition

There is an equivalence of 2-categories

CombModCat[QuillenEquivs 1]PresentableDerivators CombModCat\big[ QuillenEquivs^{-1} \big] \;\simeq\; PresentableDerivators

between the homotopy 2-category of combinatorial model categories (Prop. ) and the 2-category of presentable derivators with left adjoint morphisms between them.

This is the statement of Renaudin 06, theorem 3.4.4.

For 𝒞\mathcal{C} a 2-category write

  1. 𝒞 1\mathcal{C}_1 for the 1-category obtained by discarding all 2-morphisms;

  2. π 0 iso(𝒞)\pi_0^{iso}(\mathcal{C}) for the 1-category obtained by identifying isomorphic 2-morphisms.

Proposition

(localization of CombModCatCombModCat at the Quillen equivalences)

The composite 1-functor

CombModCat 1π 0 iso(CombModCat)π 0 iso(γ)π 0 iso(CombModCat[QuillenEquivs 1]) CombModCat_1 \longrightarrow \pi_0^{iso}(CombModCat) \overset{\pi_0^{iso}(\gamma)}{\longrightarrow} \pi_0^{iso}( CombModCat[QuillenEquivs^{-1}] )

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

Ho(CombModCat)CombModCat 1[QuillenEquivs 1]π 0 iso(CombModCat[QuillenEquivs 1]). Ho(CombModCat) \;\coloneqq\; CombModCat_1\big[ QuillenEquivs^{-1} \big] \simeq \pi_0^{iso}( CombModCat[QuillenEquivs^{-1}] ) \,.

Moreover, this localization inverts precisely (only) the Quillen equivalences.

This is the statement of Renaudin 06, cor. 2.3.8 with prop. 2.3.4.

Corollary

There is an equivalence of categories

Ho(CombModCat)Ho(PresentableDerivators) Ho(CombModCat) \;\simeq\; Ho(PresentableDerivators)

between the homotopy category of combinatorial model categories and that of presentable derivators with left adjoint morphisms between them.

Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible reflective localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.

A\phantom{A}(n,r)-categoriesA\phantom{A}A\phantom{A}toposesA\phantom{A}locally presentableloc finitely preslocalization theoremfree cocompletionaccessible
(0,1)-category theorylocalessuplatticealgebraic latticesPorst’s theorempowersetposet
category theorytoposeslocally presentable categorieslocally finitely presentable categoriesAdámek-Rosický‘s theorempresheaf categoryaccessible categories
model category theorymodel toposescombinatorial model categoriesDugger's theoremglobal model structures on simplicial presheavesn/a
(∞,1)-category theory(∞,1)-toposeslocally presentable (∞,1)-categoriesSimpson’s theorem(∞,1)-presheaf (∞,1)-categoriesaccessible (∞,1)-categories

References

Last revised on July 16, 2018 at 14:13:13. See the history of this page for a list of all contributions to it.