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 ∞-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
Sure! Here is an abstract on (co)Cartesian fibrations: We define the (co)Cartesian fibrations for ordinary categories and then extend the definition to infinity categories.
6. Straigthening-Unstraigthening (3.3)
Rafah Hajjar Muñoz 3/3/25
We go through the Straigthening-Unstraigthening construction. Notes available here.
7. Yoneda Lemma (4.1, 4.2)
Yuyuan Luo 3/10/25
First, I give the notions of initial and terminal objects of ∞-categories. Following this, I construct the mapping space functor mapc(-,-) for an ∞-category C via the twisted arrow construction. Finally, I state and sketch the proof the the Yoneda lemma. Notes available here .
8. (Co)Limits (4.3, 4.4)
Vidhu Maneka Adhihetty 3/24/25
We will define (co)limits for infinity categories in analogy with ordinary categories. Then, we state and prove some results for (co)limits on infinity categories which strengthens this analogy. We will aim to conclude with a discussion of Kan extensions in the context of infinity categories. Notes available here.
9. Adjunction between Infinity Categories (5.1, 5.2)
Hechen Hu 3/31/25
I'll start by defining adjunction between infinity categories and sketch its connection to adjunction for ordinary categories. Then I'll state theorems relating to the existence of adjoints. Finally, I'll discuss an interesting adjoint functor related to defining the K-theory of symmetric monoidal categories.
10. Presentable Infinity Categories
Carlos Andrés Alvarado Álvarez 4/7/25
We give a definition of presentable infinity categories and their tensor product. As an application we build the category of spectra with the smash product. This exposition follows Sheaves on Manifolds by Krause, Nikolaus, Putzsuck.
11. Spectra and the Stable Category
Carlos Andrés Alvarado Álvarez4/14/25
We go over the definition of a stable category and look at the category of spectra. We will see examples of different spectra, and higher algebraic phenomena that are not seen in the classical setting.
12. Derived Algebraic Geometry
Rafah Hajjar Muñoz 4/21/25
The theory of derived algebraic geometry aims to provide a setting in which geometrically bad situations (such as singularities, nontransversal intersections, quotients by nonfree actions…) can be treated the same way as their smooth counterparts. In this talk, we will give an overview of DAG and introduce the formalism of simplicial commutative rings and derived schemes, which one can think of as a simplicial resolution of a scheme by smooth ones.
13. TBD
John Smith 4/28/25
Abstract.