Infinity Categories (Spring 2022)

We will continue our exploration of infinity categories from last semester and will investigate applications of the theory, particularly to derived algebraic geometry. Our ambitious goal is to shore up our foundation in the fundamentals following [H], to learn derived algebraic geometry from [L], and then to understand the results of [B]. However, we are open to focusing on a subset of this material in greater depth. Applications of infinity category theory to other fields may also be covered, depending on participant interest.


January 31
Amal Mattoo
Organizational and Introductory Meeting
We will review the basics of infinity categories and provide an overview of the topics in this seminar. Suggestions for topics or volunteers for talks are welcome!
February 7
Amal Mattoo
Crash Course on Model Categories
We will define model categories and prove some of their key properties. Using the apparatus of cofibrantly generated model categories, we will define model structures on the categories of simplicial sets and topological spaces. Then we will examine the homotopy category of a model category and explore Quillen adjunctions and equivalences, culminating in a proof that simplicial sets and topological spaces admit equivalent homotopy categories. We aim to prove most of these results, though a few proofs will be cited from Hovey and Hinich.
February 14
Amal Mattoo
Model Categories and Beyond
We will continue with last week's material, which we did not finish covering—i.e., the homotopy theory of model categories and Quillen adjunctions and equivalences. We will examine applications including homotopy (co)limits and the derived category of an abelian category.
February 21
No meeting (Presidents' Day)
February 28
Amal Mattoo
More Models of Infinity Categories
We define models of infinity categories as simplicial categories and as complete Segal spaces. We tie these to our earlier work by introducing concepts including Dwyer-Kan localization and the Reedy model structure. We will roughly follow Sections 5 and 6 of Hinich.
March 7
Amal Mattoo
Infinity Categorical Yoneda and Universal Constructions
We construct the analogue of the Yoneda embedding for infinity categories using the model of Complete Segal Spaces. To do so, we will build the theory of left fibrations. We can then define limits, colimits, and adjoints for infinity categories. We will roughly follow Sections 7 and 8 of Hinich.
March 14
No meeting (Spring Break)
March 21
Amal Mattoo
General Infinity Categories
Zoom Link.
We wrap up our discussion of general infinity categories, now employing language independent of a choice of model. We first use Quillen adjunction to discuss maximal subspaces and infinity localization. Then we introduce cocartesian fibrations, which generalize left fibrations, and provide another approach to (co)limits. We roughly follow the end of Section 8 and Section 9 of Hinich.
March 28
Amal Mattoo
Introduction to Derived Algebraic Geometry
Turning away from general infinity categories, we begin our exploration of derived algebraic geometry. We first present a brief runthrough of stable infinity categories, particularly A-infinity and E-infinity rings. Then we will define and build up some theory on simplicial rings and derived schemes. We follow Khan and Lurie.
April 4
Amal Mattoo
Derived Modules and Quasi-coherent Sheaves
We provide some more detail on modules over simplicial rings and their properties. Then we define and present results on quasi-coherent sheaves over derived schemes, in particular descent. We again follow Khan and Lurie.
April 11
Kevin Chang
Some Features of Derived Algebraic Geometry
In this talk, I will discuss some advantages of derived schemes over ordinary schemes. We will see how Serre's intersection formula, flat base change, and the cotangent complex arise naturally and are enhanced in the derived setting. Lecture Notes.
April 18
Amal Mattoo
Bhatt's Algebrization and Tannaka Duality
We dive into the Bhatt paper, which uses infinity-categorical derived algebraic geometry to prove concrete results about algebraic spaces and schemes. We build up a bit more general theory of derived algebraic geometry and provide a brief introduction to algebraic spaces. Then we present a powerful result giving an equivalence between hom sets and functor categories. From there we state and outline the proofs of results related to formal points, gluing, and products.
April 25
Roy Magen
Some Basic Stuff about Motivic Homotopy Theory
Zoom Link.
I will present fundamental definitions and explain basic ideas from unstable motivic homotopy theory. We will see how to prove a "purity" result whose proof uses A1-homotopy and deformation to the normal cone to replace the tubular neighbourhood theorem from topology. If time permits, we will also discuss the basic ideas of algebraic K-theory (such as Bott periodicity) and algebraic cobordism from a motivic homotopy perspective, or explain some fundamentals of stable motivic homotopy theory. Main Reference: Antieau and Elmanto. Notes.
May 2
Various speakers
Derived Categories, Anabelian Geometry, and More!
We will have an informal discussion sharing what we have learned at recent conferences in Cornell, Georgia, Paris, and Cambridge.