  We use the terms "$\infty$-categories" and "$\infty$-functors" to mean the
objects and morphisms in an "$\infty$-cosmos." Quasi-categories, Segal
categories, complete Segal spaces, naturally marked simplicial sets, iterated
complete Segal spaces, $\theta_n$-spaces, and fibered versions of each of these
are all $\infty$-categories in this sense. We show that the basic category
theory of $\infty$-categories and $\infty$-functors can be developed from the
axioms of an $\infty$-cosmos; indeed, most of the work is internal to a strict
2-category of $\infty$-categories, $\infty$-functors, and natural
transformations. In the $\infty$-cosmos of quasi-categories, we recapture
precisely the same theory developed by Joyal and Lurie, although in most cases
our definitions, which are 2-categorical rather than combinatorial in nature,
present a new incarnation of the standard concepts.
  In the first lecture, we define an $\infty$-cosmos and introduce its
"homotopy 2-category," using formal category theory to define and study
equivalences and adjunctions between $\infty$-categories. In the second
lecture, we study (co)limits of diagrams taking values in an $\infty$-category
and the relationship between (co)limits and adjunctions. In the third lecture,
we introduce comma $\infty$-categories, which are used to encode the universal
properties of (co)limits and adjointness and prove "model independence"
results. In the fourth lecture, we introduce (co)cartesian fibrations, describe
the calculus of "modules" between $\infty$-categories, and use this framework
to prove the Yoneda lemma and develop the theory of pointwise Kan extensions of
$\infty$-functors.

Category Theory;Algebraic Topology