# Categorical notions involving $\ell_p$ spaces.

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 $\leftarrow$ ???

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