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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 left proper 1 combinatorial model categories.

Remark

(local presentation of combinatorial model categories)

By Dugger's theorem, we may choose for every 𝒞CombModCat\mathcal{C} \in CombModCat an sSet-category 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

Beware that, for the time being, the entry above is referring to the numbering in the arXiv version of Renaudin 2006, which differs from that in the published version.


  1. The condition of left properness does not appear in the arXiv version of Renaudin 2006, but is added in the published version. By Dugger's theorem (see here) every combinatorial model category is Quillen equivalent to a left proper one, but it is not immediate that every zig-zag of Quillen equivalences between left proper combinatorial model categories may be taken to pass through only left proper ones.

Last revised on August 30, 2021 at 14:49:54. See the history of this page for a list of all contributions to it.