Matan Prasma: Few steps towards higher group theory

In the first hour  we will see how when one weakens the unit, associativity and inverse axioms of a topological group to hold only up to coherent homotopy one gets precisely the notion of a loop space.  We will then describe few basic properties and constructions of loop spaces, which will be used in the second hour.  If time permits, we will discuss the infinity-categorical point of view.

In the second hour, viewing loop spaces as higher analogs of groups, we will describe a partial dictionary which translates classical group-theoretic notions and assertions to their counterparts in (the infinity category of) infinity-groups.  Examples of such are the isomorphism theorems, group extensions and Sylow theorems.  If time permits we will describe few applications of that approach. This is a joint work with T. Schlank.