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You should begin with the projective model structure on the category of small simplicial functors from $M^{op}$ to simplicial sets due to Chorny and Dwyer http://arxiv.org/abs/math/0607117. This model …

You can consider the marked simplicial set $(N(\mathcal{C}),\mathcal{W})$, where $N$ is the usual nerve functor and $\mathcal{W}$ is the class of weak equivalences in your model category $\mathcal{C}$ …

Even if the dual $M^{\mathrm{op}}$ of your original simplicial model category $M$ is combinatorial, so that its associated $\infty$-category $\mathcal{M}^{\mathrm{op}}$ is presentable, it is not true …

Here is another family of examples of non-cofibrantly generated model categories which admit left Bousfield localization with respect to certain classes of maps. Let $C$ be a proper simplicial model c …

If you believe Vopěnka's principle then any cofibrantly generated model category is Quillen equivalent to a combinatorial one and thus its underlying $\infty$-category is presentable. It follows that …

The following paper by Tom Leinster http://arxiv.org/abs/math/0002180 defines for any symmetric monoidal model category $M$ and any (symmetric) operad $P$ (in $Set$) the notion of an $\infty$-algebr …

There was a mistake in an earlier version of the paper that you mention. If you define $\pi_0$-fibrations and $\pi_0$-equivalences the way that you did, they do not give the structure of a category o …

Call a model structure on $Top$ (the category of topological spaces) standard, if the weak equivalences are the weak homotopy equivalences. In this nLab page, two standard model structures on $Top$ ar …