  For $K$ a field, a Wedderburn $K$-linear category is a $K$-linear category
$\sA$ whose radical $\sR$ is locally nilpotent and such that $\bar
\sA:=\sA/\sR$ is semi-simple and remains so after any extension of scalars. We
prove existence and uniqueness results for sections of the projection $\sA\to
\bar\sA$, in the vein of the theorems of Wedderburn. There are two such
results: one in the general case and one when $\sA$ has a monoidal structure
for which $\sR$ is a monoidal ideal. The latter applies notably to Tannakian
categories over a field of characteristic zero, and we get a generalisation of
the Jacobson-Morozov theorem: the existence of a pro-reductive envelope
$\Pred(G)$ associated to any affine group scheme $G$ over $K$. Other
applications are given in this paper as well as in a forthcoming one on
motives.
