  We study the problem of extension and lifting of operators belonging to
certain operator ideals, as well as that of their associated polynomials and
holomorphic functions. Our results provide a characterization of
$\mathcal{L}_1$ and $\mathcal{L}_{\infty}$-spaces that includes and extends
those of Lindenstrauss-Rosenthal \cite{LR} using compact operators and
Gonz\'{a}lez-Guti\'{e}rrez \cite{GG} using compact polynomials. We display
several examples to show the difference between extending and lifting compact
(resp. weakly compact, unconditionally convergent, separable and Rosenthal)
operators to operators of the same type. Finally, we show the previous results
in a homological perspective, which helps the interested reader to understand
the motivations and nature of the results presented.
