The process of defining what an infinity-category is becomes less recursive once one settles on a concrete definition of an $(\infty,1)-$category, intuitively a higher category with an infinite number of levels of morphisms, where $k$- morphisms are invertible for all $k \geq 2$. Four different models have arisen for this structure. They are:

Simplicially enriched categories (confusingly called a simplicial category) Complete Segal spaces Segal categories Quasi-categories (this is the definition used by Lurie in HTT)

These four definitions turn out to be equivalent in the sense that there are model category structures on the categories of each of these objects that are all Quillen equivalent