  For any Lie group $G$, we construct a $G$-equivariant analogue of symplectic
capacities and give examples when $G = \mathbb{T}^k\times\mathbb{R}^{d-k}$, in
which case the capacity is an invariant of integrable systems. Then we study
the continuity of these capacities, using the natural topologies on the
symplectic $G$-categories on which they are defined.
