# nLab Ho(CombModCat)

Contents

model category

## Model structures

for ∞-groupoids

### for $(\infty,1)$-sheaves / $\infty$-stacks

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

(∞,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 $CombModCat$ 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 $\mathcal{C} \in CombModCat$ a simplicial set $\mathcal{S}$ 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)

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

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.

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 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