Why are $\infty$-categories homotopy theories? The following phrase comes up in a few papers. 

An $\infty$-category is a homotopy theory

What does this mean?  and how does one make this precise? 

What I know: 


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*from a homotopical category, $(C,W)$  we can form an $\infty$-category $L(C,W)$ via hammock localization. 


But for one to really say that we can use $\infty$-category to replace study of homotopy theory shouldn't we show


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*The map above : "homotopical categories" -> "$\infty$-category" is fully faithful and essential surjective in some sense?

 A: "Precise" is a difficult thing to make this, because the point of such an claim is that "homotopy theory" is a non-precise notion made precise by the concept "$\infty$-category." Why? Well, a homotopy theory is something which should have objects equipped naturally with mapping spaces and a notion of equivalence "up to homotopy" and homotopy limits and colimits. All these are essentially native concepts to $\infty$-category theory. In contrast, they're very difficult to get one's hands on with the notion of homotopical category-it strikes me as quite unnatural to try to formalize the phrase "homotopy theory" as meaning "a homotopical category!" Rather, I think it's a deep and highly non-obvious theorem that a homotopical category actually contains all the information of a homotopy theory, via the simplicial localization to which you refer. Indeed, $\infty$-categories are equivalent to homotopical categories in an appropriate sense, due to a theorem of Barwick and Kan. 
The main other candidate for a formalization of "a homotopy theory" has been Quillen's model categories. However, these are neither as flexible nor as general as $\infty$-categories. It is a common attitude by now that model categories can be the best setting for making certain concrete homotopical computations, but that they, or even their generalizations to fibration or cofibration categories, are not the ideal notion of "a homotopy theory." 
One more contender as a formalization of "a homotopy theory" is the notion of derivator, which was actually introduced under the name homotopy theory by one of its inventors, Alex Heller. Derivators have homotopy limits and colimits built right in, but the mapping spaces are not as accessible, which is a significant weakness. 
