Example 15.8 (0MLE) · ScalingStacks

Example 15.8. Suppose that $\mathcal{M}$ is a model category satisfying the following list of conditions.

(M.1)

The class of weak equivalences of $\mathcal{M}$ are closed under filtered colimits.
(M.2)

Every monomorphism of $\mathcal{M}$ is a cofibration.
(M.3)

For any object $Y$ of $\mathcal{M}$ , the functor $X\mapsto X\times Y$ preserves colimits.
(M.4)

For any cofibrations $f\colon X\to Y$ and $f^{\prime}\colon X^{\prime}\to Y^{\prime}$ , the pushout product

$f\Box f^{\prime}\colon(X\times Y^{\prime})\cup^{(X\times X^{\prime})}(Y\times X^{\prime})\to Y\times Y^{\prime}$

is a cofibration that is trivial if either $f$ or $f^{\prime}$ is.
(M.5)

The $\infty$ -category $\mathrm{N}^{\mathrm{H}}\mathcal{M}$ is a homotopy theory of $(\infty,n-1)$ -categories.

Work of Bergner [12] and Lurie [29], combined with 14.6 above, shows that each of the following is an example of a model category of $(\infty,n)$ -categories:

•

the projective or (equivalently) the injective model category [29, 2.2.16, 2.3.1, 2.3.9] $\Seg_{\mathcal{M}}$ of $\mathcal{M}$ -enriched preSegal categories, and
•

the model category [28, A.3.2] $\cat_{\mathcal{M}}$ of categories enriched in $\mathcal{M}$ .

Moreover, following Simpson [37] the injective (aka Reedy) model category of Segal $(n-1)$ -categories [22, 32, 37] satisfies conditions (M.1-4); indeed, the most difficult of these to verify is (M.4), which Simpson does in [37, Th. 19.3.2 (using Corollary 17.2.6)].