Crossed modules, graded categories, and homotopy invariants of maps

In topology, groups serve as algebraic models for 1-types, which are spaces with vanishing second and higher homotopy groups. Crossed modules, on the other hand, generalize groups and are useful for modeling 2-types. In this talk, we will provide an overview of crossed modules and their properties, focusing on their relationship to topology. We will then introduce crossed module graded categories, which generalize group graded categories, and demonstrate how they can be used to construct homotopy invariants of maps from closed oriented 3-manifolds to 2-types. This is joint work with Alexis Virelizier.