nLab model structure on sections

Definitions

Sections

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.

Projective and injective model structures

Theorem

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:

1. A projective model structure in which a morphism $X\to Y$ is a fibration or weak equivalence if each $f_k : X_k\to Y_k$ is such in $F_k$.
2. An injective model structure in which a morphism $X\to Y$ is a cofibration or weak equivalence if each $f_k : X_k\to Y_k$ is such in $F_k$.

Proof

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:

Theorem

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.

Proof

This is Barwick, 4.38.

Reedy model structures

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.

Properties

• 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.

Related pages

• The dual construction is the Grothendieck construction for model categories (a lax colimit).

References

• 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