  In functional analysis there are several reasonable approaches to the notion
of a projective module. We show that a certain general-categorical framework
contains, as particular cases, all known versions. In this scheme, the notion
of a free object comes to the forefront, and in the best of categories, called
freedom-loving, all projective objects are exactly retracts of free objects. We
concentrate on the so-called metric version of projectivity and characterize
metrically free `classical', as well as quantum (= operator) normed modules.
Hitherto known the so-called extreme projectivity turns out to be, speaking
informally, a kind of `asymptotically metric projectivity'.
  Besides, we answer the following concrete question: what can be said about
metrically projective modules in the simplest case of normed spaces? We prove
that metrically projective normed spaces are exactly $l_1^0(M)$, the subspaces
of $l_1(M)$, where $M$ is a set, consisting of finitely supported functions.
Thus in this case the projectivity coincides with the freedom.
