turkmath.org

A well-known fact is that groups are algebraic models for 1-types. Generalizing groups, crossed modules model 2-types. In this talk, I will introduce the notion of a crossed module graded fusion category which generalizes that of a fusion category graded by a group. Then,using such categories, I will construct a 3-dimensional state-sum homotopy quantum field theory (HQFT) with a 2-type target. Such an HQFT associates a scalar to a map from a closed oriented 3-manifold to the fixed 2- type. Moreover, this scalar is invariant under homotopies. This HQFT generalizes the state-sum Turaev-Virelizier HQFT with an aspherical target. This is joint work with Alexis Virelizier.