Introduction to infinity-categories

The aim of this course will be to study the approach to higher category theory via the theory of infinity-categories (aka. quasi-categories) as expounded in the voluminous works of Joyal, Lurie and others. Following the tremendous impact of ordinary category theory, the scope of a theory of higher categories is to provide appropriate foundations and a versatile collection of methods in order to study the type of "categories with morphisms of arbitrarily high dimension" that emerge in topology and algebra with ever greater homotopical intensity. In the course we will discuss thoroughly the basic theory of the Joyal model category, review the connections with homotopical algebra and discuss some of the many applications of the theory. PREREQUISITES: A solid background in category theory and topology will be necessary. The subject is rooted in homotopy theory so familiarity with algebraic topology or homological algebra as well as some basic model category theory is highly desirable. FORMAT: Higher category theory contains and transcends ordinary category theory, often in highly non-trivial ways. So its breadth reaches at least as far as ordinary category theory - which is arguably far. As a consequence, revisiting ordinary notions from a higher viewpoint requires a grand amount of foundational reworking. Thus, there is an internal antagonism between the slow-paced thorough local development and the global powerful perspective of the theory. To counteract this issue, the course will proceed in two parallel streams: - The first one, which will take place twice a week with an exercises/short presentations session every two weeks, will develop the theory carefully from the beginning, while at the same time: - the second one will take place once a week and will present more advanced topics in the subject which should be possible to follow in parallel but whose full grasp will anticipate the development of the basic theory.