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Ho(CombModCat)

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Model category theory

model category

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Producing new model structures

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Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

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Introductions

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(,1)(\infty,1)-Category theory

Contents

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 Pr(∞,1)Cat, the (∞,1)-category of locally presentable (∞,1)-categories and (∞,1)-colimit-preserving (∞,1)-functors between them.

A proof for this statement, not just for homotopy categories but for the full homotopy theories ( ( , 1 ) (\infty,1) -categories), is now claimed in Pavlov 2021.

An anlogous equivalence, but with presentable derivators and just at the level of homotopy 2-categories, is due to Renaudin 06, see Corollary below.

Definition and 2-localization

Definition

(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. ΔModCat\Delta ModCat for the 2-category whose objects are simplicial model categories ( sSet Qh sSet_{Qh} -enriched model categories), whose 1-morphisms are simplicial left Quillen functors and 2-morphisms are natural transformations.

  3. CombModCatModCatCombModCat \subset ModCat and ΔCombModCaΔModCa\Delta CombModCa \subset \Delta ModCa for the full sub-2-categories on the left proper combinatorial model categories,

  4. LPropCombModCatCombModCatLPropCombModCat \subset CombModCat and LPropCombModCatCombModCatLPropCombModCat \subset CombModCat for the further full sub-2-categories on the left proper combinatorial model categories.

Remark

(local presentation of combinatorial model categories)
By Dugger's theorem, we may choose for every 𝒞CombModCat\mathcal{C} \in CombModCat (Def. ) 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}. The latter is still a combinatorial model category but is also a left proper simplicial model category.

Proposition

(the homotopy 2-category of combinatorial model categories)

The 2-localization of a 2-category

LPropCombModCat[QuillenEquivs 1] LPropCombModCat\big[QuillenEquivs^{-1}\big]

of the 2-category of left proper] [[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

LPropCombModCat[QuillenEquivs 1](𝒞,𝒟)ModCat(𝒞 p,𝒟 p)[{QuillenHomotopies} 1] LPropCombModCat\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 2.3.2.1

Relation to derivators

Proposition

There is an equivalence of 2-categories

LPropCombModCat[QuillenEquivs 1]PresentableDerivators LPropCombModCat\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

LPropCombModCat 1π 0 iso(CombModCat)π 0 iso(γ)π 0 iso(CombModCat[QuillenEquivs 1]) LPropCombModCat_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

The equivalence of the homotopy 2-category of combinatorial model categories with that of presentable derivators is due to:

The equivalence of the full homotopy theory (in particular the homotopy 2-category) of combinatorial model categories with presentable \infty -categories is due to


  1. The condition of left properness does not appear in the arXiv version of Renaudin 2006, but is added in the published version. While Dugger's theorem (Rem. ) ensures that every combinatorial model category is Quillen equivalent to a left proper one, 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 October 14, 2021 at 06:44:52. See the history of this page for a list of all contributions to it.