In this lecture we introduce the notion of a quasi-category. We then prove that we can build a homotopy category out of every quasi-category.

In order to construct such a category, we use the lifting conditions inherent to a quasi-category, to show that quasi-categories have a notion of 

• objects, 
• morphisms, 
• a homotopy of morphisms (which we prove to be an equivalence relation)
• composition, which we prove to be well-defined, associative and unital up to homotopy

Finally, we observed that the notion of homotopy category gives us a functor from quasi-categories to categories.