turkmath.org

There is a way of writing an algebraic structure (a group or an algebra or a category) as a product of two substructures. This is known as a distributive law, and also as a factorization system. After giving examples, I am going to introduce crossed simplicial groups. Crossed simplicial groups are defined by a distributive law between the simplicial category $\Delta$ and a suitable collection of groups. My main aim is to explain how one can extend the notion of 'simplicial homotopy' to crossed simplicial groups. We'll end with a very interesting example coming from Leibniz algebras that are non-skew-symmetric analogues of Lie algebras.