  We introduce a new notion of embeddability between Banach spaces. By studying
the classical Mazur map, we show that it is strictly weaker than the notion of
coarse embeddability. We use the techniques from metric cotype introduced by M.
Mendel and A. Naor to prove results about cotype preservation and complete our
study of embeddability between $\ell_p$ spaces. We confront our notion with
nonlinear invariants introduced by N. Kalton, which are defined in terms of
concentration properties for Lipschitz maps defined on countably branching
Hamming or interlaced graphs. Finally, we address the problem of the
embeddability into $\ell_\infty$.
