nLab Ho(CombModCat)

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 $CombModCat$ of combinatorial model categories. Due to Dugger's theorem, it should be true that

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

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 ($(\infty,1)$-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. $ModCat$ for the 2-category whose objects are model categories, whose 1-morphisms are left Quillen functors and 2-morphisms are natural transformations.

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

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

4. $LPropCombModCat \subset CombModCat$ and $LPropCombModCat \subset CombModCat$ 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 $\mathcal{C} \in CombModCat$ (Def. ) an sSet-category $\mathcal{S}$ 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 $\mathcal{S}$. 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

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

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

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

of the category of left Quillen functors and natural transformations between local presentations $\mathcal{C}^p$ 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 2.3.2.¹

Relation to derivators

Proposition

There is an equivalence of 2-categories

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

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

1. $\mathcal{C}_1$ 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

(localization of $CombModCat$ at the Quillen equivalences)

The composite 1-functor

$$LPropCombModCat_1 \longrightarrow \pi_0^{iso}(CombModCat) \overset{\pi_0^{iso}(\gamma)}{\longrightarrow} \pi_0^{iso}( CombModCat[QuillenEquivs^{-1}] )$$

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

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

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

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

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

Related concepts

• formal $\infty$-category theory

• 2-model category

• Ho(Cat)

• double category of model categories

• Quillen adjoint triple

• PrCat, Pr(∞,1)Cat

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.

$\phantom{A}$(n,r)-categories$\phantom{A}$	$\phantom{A}$toposes$\phantom{A}$	locally presentable	loc finitely pres	localization theorem	free cocompletion	accessible
(0,1)-category theory	locales	suplattice	algebraic lattices	Porst’s theorem	powerset	poset
category theory	toposes	locally presentable categories	locally finitely presentable categories	Adámek-Rosický‘s theorem	presheaf category	accessible categories
model category theory	model toposes	combinatorial model categories		Dugger's theorem	global model structures on simplicial presheaves	n/a
(∞,1)-category theory	(∞,1)-toposes	locally presentable (∞,1)-categories		Simpson’s theorem	(∞,1)-presheaf (∞,1)-categories	accessible (∞,1)-categories

References

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

• Olivier Renaudin, Plongement de certaines théories homotopiques de Quillen dans les dérivateurs, Journal of Pure and Applied Algebra Volume 213, Issue 10, October 2009, Pages 1916-1935

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

  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.

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

• Dmitri Pavlov, Combinatorial model categories are equivalent to presentable quasicategories (arXiv:2110.04679)

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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. ↩