In earlier work we used the double of an ideal to define in integral closure terms a notion of infinitesimal Lipschitz equisingularity for hypersurfaces. We describe different ways of extending this notion to general spaces, and show the most restrictive version is a generic condition. We then show that this condition is necessary for strong bi-Lipschitz equisingularity, a notion due to Fernandes and Ruas. Hence our condition gives a necessary condition for a smooth subset of an analytic set to be a stratum in a Mostowski stratification of the set.