Reading Seminar on Higher Category Theory, Fall 2016

Essential information

Topics

The outline of topics is available here.

Schedule

Date Speaker Title and abstract

Sep 15

 Chris Kapulkin

Motivation and Basic Concepts

 I will present the motivation to study higher category theory coming from different areas (homotopy theory, TQFTs, derived algebraic geometry) and introduce basic concepts of quasicategory theory, such as: homotopies and equivalences, proving some of their properties.

Sep 22

 Dan Christensen The Joyal model structure on simplicial sets

I will discuss the class of "weak categorical equivalences" between simplicial sets, which form the weak equivalences in the Joyal model structure, whose fibrant objects are the quasicategories. [notes]

Sep 29

 Karol Szumiło The Joyal model structure on simplicial sets II

I will construct the Joyal model structure.

Oct 6

 Alex Rolle Coherent Nerve and Simplicial Localization

This talk will introduce two new models of infinity categories (simplicially-enriched categories and relative categories), and relate them to quasicategories. [notes]

Oct 13

 Aji Dhillon Mapping spaces in higher categories (I)

The goal of these two talks is to introduce models for mapping spaces in higher categories and show that they are all equivalent. In the process we will discuss straightening and unstraightening, infinity analogue of the Grothendieck construction. We will conclude with a discussion of cartesian fibrations. [notes]

Oct 20

 Aji Dhillon Mapping spaces in higher categories (II)

The goal of these two talks is to introduce models for mapping spaces in higher categories and show that they are all equivalent. In the process we will discuss straightening and unstraightening, infinity analogue of the Grothendieck construction. We will conclude with a discussion of cartesian fibrations. [notes]

Oct 27

 Dinesh Valluri Joins, slices, and limits in quasicategories

In this talk we will introduce the notions of join, slice, (co)limits in the context of ∞-categories. We will also discuss some basic properties relevant to these constructions. [notes]

Nov 3

 Luis Scoccola Adjoint functors between quasicategories

We will generalize the concept of adjoint functors to the theory of quasicategories. [notes]

Nov 10

 Luis Scoccola Yoneda embedding for quasicategories

We will discuss some basic properties of the quasicategory of spaces, presenting in particular the quasicategorical analog of the Yoneda embedding. [notes]

Nov 17

 Marco Vergura Complete Segal Spaces

We will introduce Complete Segal spaces and prove they describe an equivalent homotopy theory to the one of quasi-categories. [notes]

Nov 24

 Marco Vergura Simplicial and relative categories

We describe how simplicial and relative categories form a model of (∞,1)-categories. [notes]

Dec 1

 James Richardson Presentable ∞-categories

I will introduce presentable quasicategories and discuss some of their properties. I will then discuss their relationship with combinatorial model categories. [notes]

Dec 8

 Pál Zsámboki

Equivalent notions of ∞-topoi

Let X be a quasicategory. Then it is an ∞-topos, if it is an accessible left exact localization of the presheaf category of a small quasicategory. We will introduce two sets of intrinsic conditions which are equivalent to being an ∞-topos:

  1. the ∞-categorical Giraud axioms, and
  2. colimits in X are universal, and it has small object classifiers for large enough regular cardinals,

and we discuss the equivalences. [notes]
For more information, please contact Chris Kapulkin.