  We exhibit Lerman's cutting procedure as a functor from the category of
manifolds-with-boundary equipped with free circle actions near the boundary,
with so-called equivariant transverse maps, to the category of manifolds and
smooth maps. We then apply the cutting procedure to differential forms that are
not necessarily symplectic, to distributions that are not necessarily contact,
and to submanifolds. We obtain an inverse functor from so-called equivariant
radial-squared blowup.
