Student seminar on higher categories

Course Description:

We want to understand the comparison problem in higher category theory, i.e. for structures which have objects, 1-morphisms, 2-morphisms, etc. Higher categories have arisen in many areas of mathematics, as well as in formalisms for quantum field theory. There are many alternative models for higher categories and the goal of this course is to understand some of them and their inter-relationships.

The end goal is to understand the Unicity Theorem (arxiv:1112.0040 [6]) which provides an axiomatization of the theory of higher categories and general tools for making these comparisons. This will be achieved using the theory of quasi-categories, in the sense that we assume our higher category models to assemble into such a structure. Quasi-categories are an especially well developed model of ($\infty$,1)-categories, where it is possible to talk about higher analogs of many 1-categorical constructions, such as limits and colimits, Kan extensions, and localizations.

Prerequisites:

Simplicial sets, their homotopy theory and relation to topological spaces. Basic category theory, limits, colimits, adjoint pairs.