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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.

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Are you aware of the work of Toen? Or the unicity theorem of Barwick and Schommer-Pries? They give axioms which any good model for $(\infty,n)$ homotopy theory should satisfy. Then they prove that all model are Quillen equivalent (and, even better, in a coherent way). In the introduction to the Unicity paper they talk a bit about why they chose to work in Quasi-categories (where the category-theoretic groundwork has been best laid), but comment that this could be done in any model. That suggests the answer to your question is Yes. –  David White Apr 12 '13 at 2:18
    
Discussion of how Dwyer-Kan style homotopy theory in Kan-complex enriched categories relates to the notions in quasi-category theory is in appendix A.3.3 of Lurie "Higher Topos Theory". A quick survey is on the nLab here: ncatlab.org/nlab/show/homotopy+Kan+extension . –  Urs Schreiber Apr 12 '13 at 5:27
    
@DavidWhite I know about the Quillen equivalences, but how does that tell me what is happening to category-theoretic constructions inside the $(\infty, 1)$-categories? –  Zhen Lin Apr 12 '13 at 7:11
    
@UrsSchreiber Thanks. Good to know that it's true in at least one pair of models! –  Zhen Lin Apr 12 '13 at 7:11

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