on chain complexes/model structure on cosimplicial abelian groups
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Introductions
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equivalences in/of $(\infty,1)$-categories
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
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.
(the 2-category of combinatorial model categories)
Write
$ModCat$ for the 2-category whose objects are model categories, whose 1-morphisms are left Quillen functors and 2-morphisms are natural transformations.
$CombModCat \subset ModCat$ for the full sub-2-category on the left proper ^{1} combinatorial model categories.
(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
from the local projective model structure on sSet-enriched presheaves over $\mathcal{S}$.
(the homotopy 2-category of combinatorial model categories)
The 2-localization of a 2-category
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
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.
There is an equivalence of 2-categories
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
$\mathcal{C}_1$ for the 1-category obtained by discarding all 2-morphisms;
$\pi_0^{iso}(\mathcal{C})$ for the 1-category obtained by identifying isomorphic 2-morphisms.
(localization of $CombModCat$ at the Quillen equivalences)
The composite 1-functor
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):
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.
There is an equivalence of categories
between the homotopy category of combinatorial model categories and that of presentable derivators with left adjoint morphisms between them.
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
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.
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 condition of left properness does not appear in the arXiv version of Renaudin 2006, but is added in the published version. By Dugger's theorem (see here) every combinatorial model category is Quillen equivalent to a left proper one, but 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 August 30, 2021 at 14:49:54. See the history of this page for a list of all contributions to it.