Introduction to higher category theory

• Lectures: Di 12:15 - 13:45 in Geom H1, Fr 14:15 - 15:45 in Geom H5
• First Lecture: Di, Oct 16
• Instructor: Tobias Dyckerhoff
• Office: Geomatikum 338
• Email: tobias.dyckerhoffATuni-hamburg de

• Tutorials: Fr 12:15 - 13:45 in Geom 142
• First Tutorial: Fr, Oct 26
• Tutor: Walker Stern
• Office: Geomatikum 340
• Email: walker.sternATuni-hamburg de

Course description

About

This course will be an introduction to higher category theory focussing on the approach via infinity-categories.

Lecture Notes

Lecture notes (until Jan 08).

Problem Sets

Problem Set 1 (due Oct 23)
Problem Set 2 (due Oct 30)
Problem Set 3 (due Nov 06)
Problem Set 4 (due Nov 13)
Problem Set 5 (due Nov 20)
Problem Set 6 (due Nov 27)
Problem Set 7 (due Dec 04)
Problem Set 8 (due Dec 11)
Problem Set 9 (due Dec 18)
Problem Set 10 (due Jan 8)
Problem Set 11 (due Jan 15)
Problem Set 12 (due Jan 22)
Problem Set 13 (due Jan 29)

Topics

• Simplicial sets and simplicial homotopy theory
• The basic language of infinity-categories
• Simplicial categories and quasi-categories
• Stable infinity-categories
• Goodwillie calculus

The main reference will be Lurie's book "Higher topos theory". More specifically, we will begin by describing the rudimentary features of the theory of infinity categories, and then move on to a more systematic study via model categories, comparison results with other models, limits and colimits, applications in the context of derived categories, and more.

Literature

• Brandenburg: Einführung in die Kategorientheorie
• Mac Lane: Categories for the working mathematician
• Weibel: An introduction to homological algebra
• Goerss-Jardine: Simplicial Homotopy Theory
• Lurie: Higher topos theory
• Lurie: Higher Algebra
• Joyal-Tierney: Quasi-categories vs Segal spaces
• This list will be updated as the course progresses.

Prerequisites

Familiarity with basic concepts from category theory and algebraic topology will be useful, but we will review those aspects that are relevant for this course.

Final Exam

Oral exam. Admission requirement: Submit solutions to 50% of the homework problems.