There are many proposed models for the theory of $(\infty, 1)$-categories and it has now been shown that many of these theories have Quillen-equivalent model categories, i.e. that they are equivalent as *homotopy* theories. But what about as *category* theories?

For example, suppose I have a homotopical category $\mathcal{C}$ and a homotopically initial object $A$ in $\mathcal{C}$, in the sense of [Dwyer, Kan, Hirschhorn, and Smith, *Homotopy limit functors on model categories and homotopical categories*]. That means, there are endofunctors $F$ and $G$ on $\mathcal{C}$ such that $\Delta A$ is weakly equivalent to $F$, $G$ is weakly equivalent to $\textrm{id}_{\mathcal{C}}$, and there is a natural transformation $\alpha : F \Rightarrow G$ such that $\alpha_A : F A \to G A$ is a weak equivalence. It is straightforward to show that $A$ descends to an initial object in $\operatorname{Ho} \mathcal{C}$, but what about in, say, the hammock localisation $L^H \mathcal{C}$ – is the hom-space $L^H \mathcal{C}(A, Z)$ contractible for every object $Z$ in $\mathcal{C}$?

More generally, how are the notions of homotopy limit/colimit, adjunction, Kan extension etc. in the different theories (where defined) related? Pointers to relevant literature would be much appreciated.

insidethe $(\infty, 1)$-categories? – Zhen Lin Apr 12 '13 at 7:11