This is the official page for the seminar on \(\infty\)-categories run by John Huerta and Diogo Freire de Andrade in the Spring Semester of the 2022/2023 academic year. Rui Peixoto took most of the notes.
Prerequisites
The seminar is designed for everyone who has been lightly exposed to Category Theory and Algebraic Topology, but we do not assume more than superficial knowledge of homotopies, functors, limits and colimits - we have designed it to be as self-contained as possible.
Seminar Structure:
The seminar will take place weekly, in a two-hour lecture format, on Wednesdays, from 14:30 to 16:30 in P5.18 (mathematics building).
Outline:
The planned structure for the lectures will be as follows:
Week 1: Introduction to the topic of (weak) \(n\)-caetgories and \((\infty,n)\)-categories.- Notes by Rui Peixoto.
Week 2: Introduction to simplicial sets and a full characterization of nerves of categories.- Notes by Rui Peixoto.
Week 3: Weak Kan complexes as \(\infty\)-categories, and suitable generalizations, the homotopy category of an \(\infty\)-category and the simplicial mapping space.- Notes by Rui Peixoto.
Week 4: Introduction to fibrations, and the construction of the functor \(\infty\)-category.- Notes (Part 1) and (part 2) by Diogo Andrade
Week 5: Proof that the function complex is an \(\infty\)-category.- Slides by Diogo Andrade.
Week 6: Introduction to Joins, Slices, the Join/Slice adjunction and limits and colimits.- Slides by Diogo Andrade.
Week 7: More on the join/slice adjunction; the definition of initial and terminal objects.- Slides by Diogo Andrade.
In the following weeks, seminar participants will present topics of their choice:
Week 8- 10 May 2023: James Macpherson on weighted joins.- Notes by James Macpherson.
Week 9- 17 May 2023: Diogo Andrade on higher Morita categories of \(\mathbb{E}_n\)-algebras- Notes by Rui Peixoto.
Week 10- 24 May 2023: Rui Peixoto on the cobordism hypothesis.- Notes by Rui Peixoto.
Week 11- 7 June 2023: Nino Scalbi on functorial field theory.- Notes by Rui Peixoto.
Week 12- 28 June 2023: Björn Gohla on Segal n-categories, a model for (∞,n)-categories.- Notes by Rui Peixoto.
- Week 13 - 26 July 2023: João Candeias on ∞-operads.
More detailed information on the lectures can be found in the page lectures. The remaining lectures are expected to be given by the participants. The topics can be chosen from the list of topics. You can email us with a choice of topic!