A well-known folklore theorem classifies 2-dimensional topological quantum field theories (TQFTs) in terms of Frobenius algebras, providing a unifying link between topology, algebra, and physics. In this talk, we explore what happens when the usual cobordism category is replaced by a category of nested cobordisms, in which 2-dimensional surfaces are equipped with embedded 1-dimensional submanifolds. We study symmetric monoidal functors out of this category and the resulting algebraic structures they encode. This talk is based on joint work with R. Hoekzema, L. Murray, N. Pacheco-Tallaj, C. Rovi, and S. Sridhar.

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).