Higher Category Theory Reading Seminar

Fall 2016

Essential information

Organized by: Dan Christensen and Chris Kapulkin.
Time: Thursday, 1 - 2:30 PM.
Location: MC 107.

Topics

The outline of the topics is available here.

Schedule

Date Speaker Title and abstract
September 15th 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.
September 22nd 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]
September 29th Karol Szumiło The Joyal model structure on simplicial sets II

I will construct the Joyal model structure.
October 6th 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]
October 13th 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]
October 20th 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]
October 27th 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]
November 3rd Luis Scoccola Adjoint functors between quasicategories

We will generalize the concept of adjoint functors to the theory of quasicategories. [notes]
November 10th 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]
November 17th 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]
November 24th Marco Vergura Simplicial and relative categories

We describe how simplicial and relative categories form a model of (∞,1)-categories. [notes]
December 1st 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]
December 8th 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, contact Chris Kapulkin.


Last updated by Chris Kapulkin on December 16th, 2016.