The goal of this seminar is to introduce the theory of Infinity Categories.
Our main application will be a construction of the stable category of spectra and the smash product with the language of infinity categories.
Location: Room 622
Meeting Time: Mondays 5:30pm EST
Reference: Introduction to Infinity-Categories by Markus Land.
1. Introduction to Infinity Categories (1.1, 1.2)
Carlos Andrés Alvarado Álvarez 01/27/25
We will introduce simplicial sets, nerves of categories and more to build up to the definition of an infinity category.
2. Joins and Fibrations (1.3, 1.4)
Carlos Andrés Alvarado Álvarez 2/3/25
We will introduce several classes of anodyne maps and fibrations that will help us solve several lifting problems. This will help us show that kan complexes are infinity groupoids, that homsets are kan complexes and that composition is unique up to contractible choice.
3. Joyal Equivalences (2.1, 2.2, 2.3)
Lisa Faulkner Valiente 2/10/25
We define conservative functors and Joyal equivalences, and prove some statements regarding the latter. In particular, we show that in the infinity category of infinity categories, a Joyal equivalence is the same as being fully faithful and essentially surjective.
4. Localizations (2.4, 2.5)
Yuyuan Luo 2/17/25
In the first part of this talk, I define localizations via a universal property, show that it is unique, and construct it. I also give some key examples. In the second part of the talk, I consider different, equivalent models for the mapping space between two objects in an oo-category. Finally, I give the definition of the coherent nerve. Notes available here .
5. (Co)Cartesian Fibrations (3.1, 3.2)
Felix Roz 2/24/25
Abstract.
6. Straigthening-Unstraigthening (3.3)
Rafah Hajjar Muñoz 3/3/25
Abstract.
7. Yoneda Lemma (4.1, 4.2)
Sofia Wood 3/10/25
Abstract.
8. (Co)Limits (4.3, 4.4)
Vidhu Maneka Adhihetty 3/24/25
Abstract.
9. Adjoint Functors (5.1, 5.2)
John Smith
Abstract.
10. Presentable Infinity Categories
John Smith
Abstract.
11. Tensor Product
John Smith
Abstract.