[[!redirects Ho(CombModCat)]] +-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Model category theory +--{: .hide} [[!include model category theory - contents]] =-- #### Homotopy theory +--{: .hide} [[!include homotopy - contents]] =-- #### $(\infty,1)$-Category theory +--{: .hide} [[!include quasi-category theory contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## 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|1-]][[category theory]] is to consider the [[homotopy category]] of the ([[very large category|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 [[equivalence of categories|equivalent]] to the [[homotopy category of an (infinity,1)-category|homotopy category]] of [[Pr(∞,1)Cat]], the [[(∞,1)-category]] of [[locally presentable (∞,1)-categories]] and [[(∞,1)-colimit]]-[[preserved limit|preserving]] [[(∞,1)-functors]] between them. A proof for this statement, not just for [[homotopy categories]] but for the full homotopy theories ([[(infinity,1)-categories|$(\infty,1)$-categories]]) now appears in [Pavlov 2025](#Pavlov25). A symmetric monoidal version of Pavlov's result also appears in [Arakawa 2026](#Arakawa26). An anlogous equivalence, but with presentable [[derivators]] and just at the level of [[homotopy 2-categories]], is due to [Renaudin 06](#Renaudin06), see Corollary \ref{EquivalenceToHoPrDer} below. ## Definition and 2-localization {#Details} \begin{definition} \label{2CategoryOfModelCategories} **([[2-category]] of [[combinatorial model category|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]]. 1. $\Delta ModCat$ for the [[2-category]] whose [[objects]] are [[simplicial model categories]] ([[classical model structure on simplicial sets|$sSet_{Qh}$]]-[[enriched model categories]]), whose [[1-morphisms]] are [[simplicial Quillen adjunction|simplicial]] [[left Quillen functors]] and [[2-morphisms]] are [[natural transformations]]. 1. $CombModCat \subset ModCat$ and $\Delta CombModCa \subset \Delta ModCa$ for the [[full sub-2-categories]] on the [[combinatorial model categories]], 1. $LPropCombModCat \subset CombModCat$ and $LPropCombModCat \subset CombModCat$ for the further [[full sub-2-categories]] on the [[left proper model categories|left proper]] [[combinatorial model categories]]. \end{definition} \begin{remark} \label{LocalPresentationOfCombinatorialModelCategories} **(local presentation of combinatorial model categories)** \linebreak By [[Dugger's theorem]], we may choose for every $\mathcal{C} \in CombModCat$ (Def. \ref{2CategoryOfModelCategories}) 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 model category|left proper]] [[simplicial model category]]. \end{remark} \begin{prop} \label{Homotopy2CategoryOf2CatOfCombinatorialModelCategories} **(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 model category|left proper] [[combinatorial model categories]] (Def. \ref{2CategoryOfModelCategories}) 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 \ref{LocalPresentationOfCombinatorialModelCategories}) at those [[natural transformation]] that on [[cofibrant objects]] have components that are [[weak equivalences]] ("Quillen homotopies"). \end{prop} This is the statement of [Renaudin 06, theorem 2.3.2](#Renaudin06).[^1] ## Relation to derivators +-- {: .num_prop} ###### 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. \ref{Homotopy2CategoryOf2CatOfCombinatorialModelCategories}) and the 2-category of presentable [[derivators]] with left adjoint morphisms between them. =-- This is the statement of [Renaudin 06, theorem 3.4.4](#Renaudin06).[^1] For $\mathcal{C}$ a [[2-category]] write 1. $\mathcal{C}_1$ for the 1-category obtained by discarding all [[2-morphisms]]; 1. $\pi_0^{iso}(\mathcal{C})$ for the [[1-category]] obtained by identifying isomorphic [[2-morphisms]]. +-- {: .num_prop} ###### 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. \ref{Homotopy2CategoryOf2CatOfCombinatorialModelCategories} 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](#Renaudin06). +-- {: .num_cor #EquivalenceToHoPrDer} ###### 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 (infinity,1)-category theory|formal $\infty$-category theory]] * [[2-model category]] * [[Ho(Cat)]] * [[2-category of model categories]] * [[double category of model categories]] * [[Quillen adjoint triple]] * [[PrCat]], [[Pr(∞,1)Cat]] * [[combinatorial simplicial model category]] [[!include categories of categories - contents]] [[!include locally presentable categories - table]] ## References {#References} The equivalence of the [[homotopy 2-category]] of [[combinatorial model categories]] with that of presentable [[derivators]] is due to: * {#Renaudin06} [[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](https://arxiv.org/abs/math/0603339), [doi:10.1016/j.jpaa.2009.02.014](https://doi.org/10.1016/j.jpaa.2009.02.014)) > Beware that, for the time being, the entry [above](#Details) is referring to the numbering in the arXiv version of [Renaudin 2006](#Renaudin06), 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 (infinity,1)-categories|presentable $\infty$-categories]] is due to * {#Pavlov25} [[Dmitri Pavlov]], *Combinatorial model categories are equivalent to presentable quasicategories*, Journal of Pure and Applied Algebra 229:2 (2025), 107860, 1–39. ([arXiv:2110.04679](https://arxiv.org/abs/2110.04679), [doi:10.1016/j.jpaa.2024.107860](https://doi.org/10.1016/j.jpaa.2024.107860)) A symmetric monoidal version of [Pavlov25]{#Pavlov25} appears in * {#Arakawa26} [[Kensuke Arakawa]], _On the equivalence of two approaches to multiplicative homotopy theories_ ([arXiv:2603.23018](https://arxiv.org/abs/2603.23018)) [^1]: The condition of left properness does not appear in the arXiv version of [Renaudin 2006](#Renaudin06), but is added in the published version. While [[Dugger's theorem]] (Rem. \ref{LocalPresentationOfCombinatorialModelCategories}) 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. [[!redirects HoCombModCat]] [[!redirects 2Ho(CombModCat)]]