# Reading Seminar on Higher Category Theory, Fall 2016

# Essential information

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

# Topics

The outline of topics is available here.

# Schedule

Date |
Speaker |
Title and abstract |

Sep 15 |
Chris Kapulkin |
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 setsI 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 III will construct the Joyal model structure. |

Oct 6 |
Alex Rolle |
Coherent Nerve and Simplicial LocalizationThis 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 quasicategoriesIn 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 quasicategoriesWe will generalize the concept of adjoint functors to the theory of quasicategories. [notes] |

Nov 10 |
Luis Scoccola |
Yoneda embedding for quasicategoriesWe 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 SpacesWe 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 categoriesWe describe how simplicial and relative categories form a model of (∞,1)-categories. [notes] |

Dec 1 |
James Richardson |
Presentable ∞-categoriesI 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 |
- the ∞-categorical Giraud axioms, and
- 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.