model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
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general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
Let $K$ be a small category and $F:K\to ModelCat$ a pseudofunctor, where $ModelCat$ is the 2-category of model categories, Quillen adjunctions pointing in the direction of their left adjoints, and mate-pairs of natural isomorphisms. This is sometimes called a Quillen presheaf (with “right” or “left” attached according to whether one is thinking about the left adjoints or the right adjoints in the Quillen adjunctions); we denote the Quillen adjunction $F(f)$ by $f_! \dashv f^*$.
The category of sections of $F$ is the category of sections $K \to \int F$ of its Grothendieck construction. Explictly, a section of $F$ consists of, for each $k\in K$ an object $X_k\in F_k$, and for each $f:\ell\to k$ in $K$ a morphism $X_\ell \to f^* X_k$ (or equivalently $f_! X_\ell \to X_k$) satisfying a functoriality condition. The category $Sect(F)$ is the lax limit of the diagram $K\to Cat$ consisting of the left Quillen functors $f_!$, and also the colax limit of the analogous diagram consisting of the right Quillen functors. Accordingly, it is locally presentable if each category $F_k$ is.
If each $F_k$ is a combinatorial model category, accessible model category, or tractable model category, then $Sect(F)$ has two model structures that are also combinatorial, accessible, or tractable respectively:
The combinatorial and tractable cases are Barwick, 2.28 and 2.30. They are a straightforward application of transferred model structures, generalizing the usual constructions of model structures on functors. The accessible case follows similarly using the accessible version of transfer.
These model structures are a presentation of the $(\infty,1)$-categorical lax limit of $F$. To present the pseudo (homotopy) limit of $F$ we have to localize them:
The above projective model structure has a left Bousfield localization, called the (projective) homotopy limit model structure, in which the fibrant objects are the projectively fibrant ones that are homotopy cartesian, i.e. for each $f:\ell\to k$ in $K$ the induced map $X_\ell \to f^*(X_k) \to f^*(R X_k)$ is a weak equivalence, where $R$ denotes a fibrant replacement.
This is Barwick, 4.38.
If each $F_k$ is a model category (not even necessarily cofibrantly generated) and $K$ is a Reedy category, then there is also a Reedy model structure on $Sect(F)$ defined analogously to the ordinary case. This is in Balzin18, together with a generalization to any “admissible model semifibration” over a Reedy category.
These model structures are presentations of the $(\infty,1)$-category of sections (i.e. (co)lax limit) of the induced functor of $(\infty,1)$-categories. For the projective model structure this is shown in Harpaz18 (hence also for the injective model structure, since they have the same weak equivalences), which considers more generally sections defined on a simplicial category. For the Reedy model structure it is shown in Balzin18, with the special case of an inverse category being in Spitzweck10.
A subcategory of the injective model structure analogous to the homotopy limit model structure is shown in Bergner10 to present the homotopy limit of the corresponding diagram of complete Segal spaces.
Clark Barwick, On left and right model categories and left and right Bousfield localizations. Homology, Homotopy and Applications, 12.2, 2010, p. 245-320. pdf, related arxiv preprint
Julia E. Bergner, Homotopy limits of model categories and more general homotopy theories, arXiv:1010.0717
Markus Spitzweck, Homotopy limits of model categories over inverse index categories, 2010, doi
Johnson, Mark W. On modified Reedy and modified projective model structures. Theory Appl. Categ. 24 (2010), No. 8, 179–208.
Yonatan Harpaz, Lax limits of model categories, pdf
Edouard Balzin, Reedy Model Structures in Families, arXiv:1803.00681
Last revised on April 20, 2023 at 16:45:16. See the history of this page for a list of all contributions to it.