Direct proofs of properties and structures of model structures for (∞, 1)-categories

There are many different models for (∞,1)-categories, each of which has an associated model category. Given these model structures, we’d like to know what additional properties they possess, for example, which are simplicial, cartesian, or left or right proper? We know several of these results, but not all; and in particular there are not always counterexamples in the literature for when a model structure does not have a desired property. Furthermore, some of these properties are known from very general results, and it would be nice to have a more concrete proof for a given model category. In this project, we’ll seek to fill in some of these gaps in our knowledge. 


Reading List: 


• J. Bergner, “A survey of (∞, 1)-categories” In: Baez, J., May, J.P. (eds) Towards Higher Categories, Vol. 152. Springer, NY. 


• P. Hirschhorn, Model Categories and their Localizations, for background on sim- plicial, monoidal, proper model categories.