Fall 2013

oo-Categories Reading Group

Wednesdays, 3 PM to 4 PM (right after class), in Room 232 Science Center. The goal of this class is for students to understand the first one or two chapters of Jacob Lurie's ``Higher Topos Theory'', and to understand the first chapter of ``Higher Algebra''.

Resources

1. Lurie, "Higher Topos Theory".
2. Lurie, "Higher Algebra".
3. Greg Friedman, "An elementary illustrated introduction to simplicial sets".
4. Paul Goerss and Rick Jardine, "Simplicial homotopy theory". Available upon request from Hiro for purposes of this class only.
5. Saunders Mac Lane, "Categories for the Working Mathematician". Available upon request from Hiro for for purposes of this class only.
6. Peter May, "Simplicial objects in algebraic topology". Available upon request from Hiro for for purposes of this class only.
7. Charles Weibel, "An Introduction to Homological Algebra". First chapter available upon request from Hiro for purposes of this class only.

Reading Assignments

1. For October 3 and October 10: Understand Section 7 and Theorem 11.4 of Goerss-Jardine, Chapter I. If we understand this, we will know what we mean by Kan complexes and why they are like topological spaces. Read the material and try to understand it so we can discuss in class. If you want a copy of this, e-mail me personally.
2. For October 17 and October 24: Read all of 1.1 in Higher Topos Theory, and also sections 1.2.1 to 1.2.11 (inclusive). Define oo-category (aka weak Kan complex, aka quasi-category). We motivate it by the example of the nerve of a strict category. How does an oo-category have Hom spaces? If we are still stuck on Goerss-Jardine, we will continue with it as needed.
3. For October 31 and November 7: Read Sections II.6 and Chapter III of Mac Lane, "Categories for the Working Mathematician". The goal is to understand limits and colimits and compute examples. What is the (co)limit of the empty diagram? Of the diagram with two objects? How about cokernels and kernels? If you need a copy of Mac Lane, e-mail me personally.
4. For November 14: Finish Chapter 1 of Higher Topos Theory. This means (if you understood everything up to here) we will know what limits and colimits are for an oo-category. Read also Section 1.5 of Weibel. Now you know what a mapping cone is in chain complexes. Brief remark about how to actually compute homotopy (co)limits: You use model structures.
5. For November 21: Read Higher Algebra, Sections 1.1 and 1.2. We will know what a stable oo-category is, and what the shift functor does.

Current Questions

1. What are the homotopy groups of BG and EG?
2. What are BG and EG?
3. What is a simplicial group?
4. How are they related to the homotopy groups of a space?
5. What are the homotopy groups of a Kan complex?
6. What is the singular chain complex?
7. What is a Kan complex?

Big Ideas

1. Our first orders of business will be:
   1. Nerves of categories are simplicial sets
   2. Kan complexes are spaces
   3. Why is homotopy equivalence important
   4. Why are model categories important

1. Sitan Chen
2. David Ding
3. Wenbo Fu
4. Ping Gao
5. Ben Kuhn
6. Rolando La Placa
7. Ben Li
8. Jake McNamara
9. Seth Neel
10. Jake Postema
11. Arpon Raksit
12. Sahana Vasudevan
13. Allen Yuan

Notes