Abstract
We construct an $(\infty,1)$-functor that takes each smooth $G$-manifold with
corners $M$ to the space of equivariant smooth $h$-cobordisms
$H_{\mathrm{Diff}}(M)$. We also give a stable analogue $H^U_{\mathrm{Diff}}(M)$
where the manifolds are stabilized with respect to representation discs. The
functor structure is subtle to construct, and relies on several new ideas. In
particular, for $G=e$, we get an $(\infty,1)$-functor structure on the smooth
$h$-cobordism space $H_{\mathrm{Diff}}(M)$. This agrees with previous
constructions as a functor to the homotopy category.