  We present a coherent definition of dagger $(\infty,n)$-category in terms of
equivariance data trivialized on parts of the category. Our main example is the
bordism higher category $\mathbf{Bord}_{n}^X$. This allows us to define a
reflection-positive topological quantum field theory to be a higher dagger
functor from $\mathbf{Bord}_{n}^X$ to some target higher dagger category
$\mathcal{C}$. Our definitions have a tunable parameter: a group $G$ acting on
the $(\infty,1)$-category $\mathbf{Cat}_{(\infty,n)}$ of
$(\infty,n)$-categories. Different choices for $G$ accommodate different
flavours of higher dagger structure; the universal choice is $G =
\operatorname{Aut}(\mathbf{Cat}_{(\infty,n)}) = (\mathbb{Z}/2\mathbb{Z})^n$,
which implements dagger involutions on all levels of morphisms. The Stratified
Cobordism Hypothesis suggests that there should be a map $\mathrm{PL}(n) \to
\operatorname{Aut}(\mathbf{AdjCat}_{(\infty,n)})$, where $\mathrm{PL}(n)$ is
the group of piecewise-linear automorphisms of $\mathbb{R}^n$ and
$\mathbf{AdjCat}_{(\infty,n)}$ the $(\infty,1)$-category of
$(\infty,n)$-categories with all adjoints; we conjecture more strongly that
$\operatorname{Aut}(\mathbf{AdjCat}_{(\infty,n)}) \cong \mathrm{PL}(n)$. Based
on this conjecture we propose a notion of dagger $(\infty,n)$-category with
unitary duality or $\mathrm{PL}(n)$-dagger category. We outline how to
construct a $\mathrm{PL}(n)$-dagger structure on the fully-extended bordism
$(\infty,n)$-category $\mathbf{Bord}_n^X$ for any stable tangential structure
$X$; our outline restricts to a rigorous construction of a coherent dagger
structure on the unextended bordism $(\infty,1)$-category
$\mathbf{Bord}_{n,n-1}^X$. The article is a report on the results of a workshop
held in Summer 2023, and is intended as a sketch of the big picture and an
invitation for more thorough development.
