  In this paper we redevelop the foundations of the category theory of
quasi-categories (also called infinity-categories) using 2-category theory. We
show that Joyal's strict 2-category of quasi-categories admits certain weak
2-limits, among them weak comma objects. We use these comma quasi-categories to
encode universal properties relevant to limits, colimits, and adjunctions and
prove the expected theorems relating these notions. These universal properties
have an alternate form as absolute lifting diagrams in the 2-category, which we
show are determined pointwise by the existence of certain initial or terminal
vertices, allowing for the easy production of examples.
  All the quasi-categorical notions introduced here are equivalent to the
established ones but our proofs are independent and more "formal". In
particular, these results generalise immediately to model categories enriched
over quasi-categories.
