I’ll describe two approaches to constructing a universal state sum. The first approach (arXiv:2104.02101) is more elementary and assumes semisimplicity. Special cases of this state sum include Turaev–Viro, Crane–Yetter, Douglas–Reutter, the Reshetikhin–Turaev Dehn surgery formula (thought of as a state sum), Brown–Arf for $\mathrm{Pin}_-$ 2-manifolds, and Dijkgraaf–Witten. The second approach (joint work with David Reutter) is more general and does not assume semisimplicity. If there’s time I’ll sketch a program to use the non-semisimple state sum to reproduce a cluster of non-semi-simple 3-manifold invariants due to many different authors (Lyubashenko, Kuperberg, Hennings, ... Geer, Gainutdinov, Patureau-Mirand, ... ).

It is well known that Frobenius algebras are in correspondence with 2-dimensional TQFTs. In this talk, we introduce Frobenius objects in any monoidal category, and in particular in the category where objects are sets and morphisms are spans of sets. We prove the existence of a simplicial set that encodes the data of the Frobenius structure in this category. This serves as a (simplicial) toy model of the Wehrheim–Woodward construction for the symplectic category. This is part of a program that intends to describe, in terms of category theory, the relationship between symplectic groupoids and topological field theory via the Poisson sigma model. Based on joint work with Rajan Mehta and Molly Keller (Rev. in Math. Phys (34) 10 (2022)), with Rajan Mehta, Adele Long and Sophia Marx (https://arxiv.org/abs/2208.14716), and ongoing work with Rajan Mehta and Walker Stern.