@Adeel Could you explain a little bit what does it mean by The model structure on the category of dg-categories "presents" an $(\infty,1)$=category DGCAT?

@QiaochuYuan Is the Dwyer-Kan model structure has a interpretation in that $(\infty,2)$-category structure?

@QiaochuYuan Could you explain a little bit what does it mean by an $\infity$-category has zero morphisms?

@DavidHill For $\mathfrak{sl}(2)$ the only thing I know is that the Verma module $M(\lambda)$, which is infinite dimensional, does not satisfy the property above. Actually I don't have many examples in head. Do you have a counter-example?