Recent work suggests a manifestly unitary construction of higher Hilbert spaces. A key desideratum for higher Hilbert spaces is that they are the local data of fully extended unitary (reflection positive) TQFTs. “Disk-like” higher categories in the sense of Morrison–Walker are defined to describe the local data of a fully extended TQFT in the absence of unitarity. In this talk, I will introduce a notion of unitary disk-like $n$-category meant to capture the local data of a fully extended unitary TQFT, briefly review 2- and 3-Hilbert spaces, and discuss the equivalence between unitary disk-like $n$-categories and $(n+1)$-Hilbert spaces in the cases $n=1$ and $n=2$.
