First of all, apologies for a somewhat vague question but let me give a try. We know what the projective objects in the category of Banach spaces are: these are precisely $\ell_1(\Gamma)$-spaces. (One the local level, which I am not going to define here, it is enough to work with $\mathscr{L}_1$-spaces.)
Similarly, we understand 1-injectivity completely (here the injective objects are $C(X)$-spaces for $X$ extremally disconnected). Again, on the local level we can work with $\mathscr{L}_\infty$-spaces. Let me make then an assignment:
$\mathscr{L}_1$-spaces $\leftarrow $ surjective operators $\leftrightarrow $ quotients
$\mathscr{L}_\infty$-spaces $\leftarrow $ isomorphic embeddings $\leftrightarrow $ closed subspaces
Can we complete this dictionary for $p\in (1,\infty)$:
$\mathscr{L}_p$-spaces $\leftarrow $ ???
