  We describe several ordinal indices that are capable of detecting, according
to various metric notions of faithfulness, the embeddability between pairs of
Polish spaces. These embeddability ranks are of theoretical interest but seem
difficult to estimate in practice. Embeddability ranks, which are easier to
estimate in practice, are embeddability ranks generated by Schauder bases.
These embeddability are inspired by the nonlinear indices \`a la Bourgain from
\cite{BLMS_FM}. In particular, we resolve a problem \cite[Problem
3.10]{BLMS_FM} regarding the necessity of additional set-theoretic axioms
regarding the main coarse universality result of \cite{BLMS_FM}.
