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

2. $CombModCat \subset ModCat$ 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 $\mathcal{C} \in CombModCat$ 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 $\mathcal{S}$.

Proposition

(the homotopy 2-category of combinatorial model categories)

The 2-localization of a 2-category

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

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

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

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

Proposition

There is an equivalence of 2-categories

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

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 $\mathcal{C}$ 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

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

induced from the 2-localization of Prop. exhibits the ordinary localization of a category of the 1-category $CombModCat$ 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

• one might expect that $Ho_2(CombModCat)$ is similarly equivalent to the full full sub-2-category of the homotopy 2-category of (∞,1)-categories on the presentable (∞,1)-categories, but this seems to remain open

• Ho(Cat)

• double category of model categories

• 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

• 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)