**Introduction to infinity-categories**Justin Noel, Georgios Raptis

**Semester**WiSe 2015 / 16

**Content / Literature / Recommended previous knowledge** 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.

**Zeit und Raum der Veranstaltung**Mo 10-12 M102 // Do 10-12 M006 // Fr 14-16 M103

**Art der Veranstaltung**Vorlesung

**Zeit und Raum der Zentralübung**Do 10-12 M006 (alle zwei Wochen)

**Zielgruppen**Master, PhD students, Postdocs

**Prüfungsbestandteile**mündliche Prüfung (30 Min.) und Übungen

**Termine und Dauer von Prüfung und erster Wiederholungsprüfung**Termin nach Absprache

**Anmeldeverfahren und Termine zu den Prüfungsbestandteilen**Per E-mail

**Liste der Module**MV, MGAGeo

**Leistungspunkte**9 LP