Notes on classification of left fibrations The quasicategory of spaces $\mathcal{S}$ classifies left fibrations analogous to the way $\mathbf{Set}$ classifies discrete left fibrations. It’s hard to find a reference where this fact is proved in detail. Here I’m making some notes on this.
Limits and colimits in the quasicategory of spaces In HTT it’s shown that $\mathcal{S}$ has all limits and colimits by proving a general result (Theorem 4.2.4.1 in HTT) relating homotopy limits and colimits in a simplicial model category to limits and colimits in the associated quasicategory. I thought it would be nice if there was a more elementary way of showing that $\mathcal{S}$ has all of these dudes; I’m in the process of writing out what I hope is such a way.