Talk series University of Vienna

Simplicial methods for higher categories: Segal-type models of weak n-categories.

Dr Simona Paoli, University of Leicester.

Lecture 1: Higher categories: Introduction and background.

The first lecture provides a broad introduction to higher categories. I will also cover some of the main techniques from category theory and simplicial homotopy theory which are needed. I will explain the multi-simplicial description of strict n-categories and n-fold categories: this is useful to build the geometric intuition around the Segal-type models.

Lecture 2: The three Segal-type models

This lecture will start with a non-technical overview of the main ideas and results regarding the Segal-type models. I will then give the precise definition of the models in dimension 2. I will define homotopically discrete n-fold categories, in preparation for the formal definition of the models for general n.

Lecture 3: The three Segal-type models and Segalic pseudo-functors

I will formally introduce the three Segal-type models in all dimensions. I will discuss the relation between weakly globular n-fold categories and a class of pseudo-functors called Segalic pseudo-functors. I will discuss the main result that the strictification of a Segalic pseudo-functor is a weakly globular n-fold category.

Lecture 4: Rigidification of weakly globular Tamsamani n-categories

The main goal of this lecture is to explain the construction of the rigidification functor from weakly globular Tamsamani n-categories to weakly globular n-fold categories.

Lecture 5: Weakly globular n-fold categories as a model of weak n-categories

In this lecture I will explain the construction of the discretization functor from weakly globular n-fold categories to Tamsamani n-categories, and the final results: the equivalence after localization of Tamsamani n-categories and weakly globular n-fold categories, exhibiting the latter as a model of weak n-categories, and the proof of the homotopy hypothesis. If time permits I will also discuss some open questions arising from this work.

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