On the categorical spectrum of topological quantum field theories

As originally suggested by Kitaev, invertible topological quantum field theories of varying dimensions should assemble into a spectrum/generalized homology theory. A candidate for such a spectrum of invertible TQFTs was proposed by Freed and Hopkins, with the defining property that (isomorphism classes of) n-dimensional invertible TQFTs are completely determined by their partition functions on closed n-manifolds.

More generally, not-necessarily-invertible TQFTs should assemble into a `categorical spectrum', an analogue of a spectrum with non-invertible cells at each level.

In this talk, I will explain that there exists a unique such categorical spectrum satisfying  a list of reasonable assumptions on the collection of (compact/very finite and discrete) TQFTs; one of these assumptions being that its invertibles agree with Freed and Hopkins' suggestion.

I will explain these assumptions, sketch how this categorical spectrum looks like in low-dimensions, outline its construction, and how it may be used to learn about gapped boundaries of anomaly theories in high dimensions.

This is based on work in progress with Theo Johnson-Freyd.