Involutions on $(\mathrm{\infty },n)$-categories of bordisms and reflection TQFTs

We construct a strict involution, that is, a strict $\mathbb{Z}\mathrm{/}2$-action, on the $(\mathrm{\infty },n)$-category ${\mathbf{B}\mathbf{o}\mathbf{r}\mathbf{d}}_n(H_n)$ of bordisms with $H_n$-structure, where $H_n$ is the symmetry type of a TQFT, using the complete $n$-fold Segal space model. Using this involution, we develop an appropriate definition of reflection TQFTs in this setting and show that they correspond to certain homotopy fixed point spaces. Finally, we apply the machinery to the example of super algebras, where, depending on the symmetry type, the homotopy fixed point data corresponds to either a stellar structure or a real structure on the algebra that the reflection theory assigns to the point.