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We use the terms "$\infty$-categories" and "$\infty$-functors" to mean theobjects and morphisms in an "$\infty$-cosmos." Quasi-categories, Segalcategories, complete Segal spaces, naturally marked simplicial sets, iteratedcomplete Segal spaces, $\theta_n$-spaces, and fibered versions of each of theseare all $\infty$-categories in this sense. We show that the basic categorytheory of $\infty$-categories and $\infty$-functors can be developed from theaxioms of an $\infty$-cosmos; indeed, most of the work is internal to a strict2-category of $\infty$-categories, $\infty$-functors, and naturaltransformations. In the $\infty$-cosmos of quasi-categories, we recaptureprecisely the same theory developed by Joyal and Lurie, although in most casesour 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 studyequivalences and adjunctions between $\infty$-categories. In the secondlecture, we study (co)limits of diagrams taking values in an $\infty$-categoryand the relationship between (co)limits and adjunctions. In the third lecture,we introduce comma $\infty$-categories, which are used to encode the universalproperties of (co)limits and adjointness and prove "model independence"results. In the fourth lecture, we introduce (co)cartesian fibrations, describethe calculus of "modules" between $\infty$-categories, and use this frameworkto prove the Yoneda lemma and develop the theory of pointwise Kan extensions of$\infty$-functors.

Infinity category theory from scratch