Trisections give a diagrammatic description of smooth 4-manifolds, similar to Heegaard splittings in dimension three. In this talk, I will describe new 4-manifold invariants defined from trisection diagrams using categorical data. The input consists of three spherical fusion categories, a semisimple bimodule category with a bimodule trace, and a pivotal functor into the category of bimodule endofunctors.

The construction works by labelling the trisection diagrams with the categorical data and evaluating them using a diagrammatic calculus for bimodule categories. The details of this procedure ensures that the result is invariant under moves on trisections yielding the same 4-manifold. This construction generalises existing Hopf algebraic trisection invariants due to Chaidez--Cotler--Cui and recovers the Crane--Yetter and Bärenz--Barrett invariants as special cases. I will outline the main ideas of the construction and briefly discuss examples and connections to TQFTs.

Based on the work 2511.19384 with Catherine Meusburger (FAU) and Fiona Torzewska (Bristol).