We develop the theory of (infty-)categories synthetically. Instead of being built out of more fundamental objects, we view (infty-)categories as a primitive objects. Instead of saying "what categories are", we introduce axiomatically "how they behave" and what one can do with them.

Synthetic category theory offers a foundation of mathematics in which the structure identity principle is available for _all_ higher categorical structures, by which we mean the ability to treat isomorphic/equivalent structures as logically equal, with all the benefits that this entails. This is drastically different from classical foundations (where the structure identity principle is inconceivable) and improves on homotopy type theory, which does offer the structure identity principle both for low-dimensional structures (groups, rings, 1-categories) and infty-groupoids, but is crucially not known (and expected to be unable) to do this for infty-categories or other infinitary categorical structures, such as infty-operads or (infty,n)-categories.

Beyond the logical advantages of the new foundations, synthetic category theory offers two practical benefits:

• It allows students to learn how to work with (infty-)categories while avoiding the notoriously complicated combinatorics of classical set based models (such as simplicial sets).
• It provides conceptual and high level arguments that can be interpreted not only for usual (higher) categories, but for example also for sheaves thereof.

In this lecture we will cover all the basic theory of (infty-)categories, including (co)limits, Kan extensions, adjunctions, Yoneda lemma, etc.

This course will also be taught in parallel in Hamburg by Lyne Moser.

• Registration for course work/examination/ECTS: FlexNow

• Successful participation in the exercise classes:

• Oral exam: Duration: 20-30 min, Date: by appointment, re-exam: Date: by appointment