Module 24-M-AL-HCAT Higher Categories

Non-official translation of the module descriptions. Only the German version is legally binding.

Students master advanced content and methods of Higher Categories, in particular they can independently carry out very complex proofs in this area requiring a higher level of mathematical expertise. They acquire the foundations of higher category theory to be able to routinely apply it in other fields of mathematics. Concretely:

• Students should be able to define central concepts of the theory (e.g. higher category, anima, cocartesian fibrations, Kan extensions) and use these in context
• Students know leading examples of the theory (e.g. animae, derived categories, diagram categories) and can use these examples to illustrate concepts and theorems, e.g. students can compute Kan extensions in simple situations.

Students will be introduced to current research questions in the field of Higher Categories. They are able to recognise and assess further development opportunities and research goals.
Furthermore, students recognise further-reaching connections to mathematical issues that have already been worked out. They can transfer and apply the knowledge and methods they have learnt so far to deeper mathematical problem areas. Students also expand their mathematical intuition as a result of more intensive study.
In combination with other in-depth modules, they will be able to write their own research papers, e.g. a master's thesis in the field of Higher Categories.
In the tutorials, students develop their ability to discuss mathematical topics and thus further prepare themselves for the requirements of the Master's module, in particular for the scientific discussion within the Master's seminar presentation and the defence of their Master's thesis.

The following advanced content of teaching from the field of Higher Categories is compulsory:

• Quasicategories and the coherent nerve
• Joyal’s fibration theory
• Grothendieck’s homotopy hypothesis
• Straightening/Unstraightening
• Ccolimits, adjunctions and Kan extensions

In addition, the following content of teaching can be covered, for example:

• Complete Segal animae
• Monoidal higher categories)
• E_1 and E_\infty-groups
• Spectra
• Operads
• Animation
• Basics of algebraic K-theory

This module prepares the content of a master's thesis.

Solid knowledge in basic ordinary category theory (functors, natural transformations, Yoneda’s lemma, (co)limits is expected;
Recommended: Basics of homotopy theory (e.g. from Algebraic Topology 2)

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