Multiple Lie theory
Multiple Lie theory is not the same as `higher structures' but it is an approach to some of the same problems.
The starting point is the observation that a double groupoid -- that is, a groupoid object in the category of groupoids -- with a suitable Lie structure, can be differentiated in both directions and so results in a second-order invariant, the double Lie algebroid.
We will describe this process, using only the most basic notions of category theory. It is remarkable how far one can go with this technique.
The talk will provide an overview of `Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids', Crelles Journal. Volume 2011, Issue 658, Pages 193–245.